The Peterson family used their sprinkler for 30 hours, while the Stewart family used theirs for 15 hours.
Let's assume that the Peterson family used their sprinkler for a certain number of hours, which we'll denote as x, and the Stewart family used their sprinkler for the remaining hours, which would be 45 - x.
The water output rate for the Peterson family's sprinkler is given as 35 L per hour. Therefore, the total water output for the Peterson family can be calculated by multiplying the water output rate (35 L/h) by the number of hours they used the sprinkler (x): 35x.
Similarly, for the Stewart family, with a water output rate of 40 L per hour, the total water output for their sprinkler is given by 40(45 - x).
According to the problem, the combined total water output for both families is 1,650 L. Therefore, we can write the equation:
35x + 40(45 - x) = 1,650.
Simplifying the equation, we get:
35x + 1,800 - 40x = 1,650,
-5x = 1,650 - 1,800,
-5x = -150.
Dividing both sides of the equation by -5, we find:
x = -150 / -5 = 30.
So, the Peterson family used their sprinkler for 30 hours, and the Stewart family used theirs for 45 - 30 = 15 hours.
Therefore, the Peterson family used their sprinkler for 30 hours, while the Stewart family used theirs for 15 hours.
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Hungry Harry is a giant ogre with an even bigger appetite. After Harry wakes up from hibernation, his daily hunger � ( � ) H(t)H, left parenthesis, t, right parenthesis (in kg kgstart text, k, g, end text of pigs) as a function of time � tt (in hours) can be modeled by a sinusoidal expression of the form � ⋅ cos ( � ⋅ � ) + � a⋅cos(b⋅t)+da, dot, cosine, left parenthesis, b, dot, t, right parenthesis, plus, d. When Harry wakes up at � = 0 t=0t, equals, 0, his hunger is at a maximum, and he desires 30 kg 30 kg30, start text, space, k, g, end text of pigs. Within 2 22 hours, his hunger subsides to its minimum, when he only desires 15 kg 15 kg15, start text, space, k, g, end text of pigs. Find � ( � ) H(t)H, left parenthesis, t, right parenthesis.
The equation for Harry's hunger in terms of time can be written as,H(t) = 7.5.cos(π.t) + 22.5
Given:Hunger of Harry as a function of time,H(t)H(t) can be modeled by a sinusoidal expression of the form,a⋅cos(b⋅t)+da⋅cos(b⋅t)+d, where Harry wakes up at t=0t=0t=0, his hunger is at a maximum, and he desires 30 kg 30 kg30, start text, space, k, g, end text of pigs.
Within 2 22 hours, his hunger subsides to its minimum, when he only desires 15 kg 15 kg15, start text, space, k, g, end text of pigs.
Therefore, the equation of the form for H(t)H(t) will be,H(t) = A.cos(B.t) + C where, A is the amplitude B is the frequency (number of cycles per unit time)C is the vertical shift (or phase shift)
Thus, the maximum and minimum hunger of Harry can be represented as,When t=0t=0t=0, Harry's hunger is at maximum, i.e., H(0)=30kgH(0)=30kg30, start text, space, k, g, end text.
When t=2t=2t=2, Harry's hunger is at the minimum, i.e., H(2)=15kgH(2)=15kg15, start text, space, k, g, end text.
According to the given formula,
H(t) = a.cos(b.t) + d ------(1)Where a is the amplitude, b is the angular frequency, d is the vertical shift.To find the value of a, subtract the minimum value from the maximum value.a = (Hmax - Hmin)/2= (30 - 15)/2= 15/2 = 7.5To find the value of b, we will use the formula,b = 2π/period = 2π/(time for one cycle)The time for one cycle is (2 - 0) = 2 hours.
As Harry's hunger cycle is a sinusoidal wave, it is periodic over a cycle of 2 hours.
Therefore, the angular frequency,b = 2π/2= π
Therefore, the equation for Harry's hunger in terms of time can be written as,H(t) = 7.5.cos(π.t) + 22.5
Answer: H(t) = 7.5.cos(π.t) + 22.5.
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In this case the linear equations are given:
A company offers two data plans for cell phones. The plan A the linear function for the
charge is given by
y=10x
where x represents the total number of megabytes. The Plan B charge is calculated using
the linear function
y = 4x + 75.
How many megabytes would a customer need to use for Plan be to be a better deal?
1) more than 12.5 megabytes
2) less than 18.75 megabytes
3)
Plan b is always a better deal because the charge per megabyte is less than in
plan A
4) More than 10 megabytes.
5) More than 20 megabytes
The customer would need to use more than 12.5 megabytes for Plan B to be a better deal. (Option 1) more than 12.5 megabytes).
To determine when Plan B would be a better deal than Plan A, we need to compare the charges for both plans based on the number of megabytes used.
Plan A is represented by the linear function y = 10x, where x represents the total number of megabytes used, and y represents the charge for the plan.
Plan B is represented by the linear function y = 4x + 75, where x represents the total number of megabytes used, and y represents the charge for the plan.
To find the point at which Plan B becomes a better deal, we need to find the x-value where the charge for Plan B is less than the charge for Plan A.
In other words, we need to find the x-value that satisfies the inequality:
4x + 75 < 10x
To solve this inequality, we subtract 4x from both sides:
75 < 6x
Then, we divide both sides by 6:
12.5 < x
Therefore, the customer would need to use more than 12.5 megabytes for Plan B to be a better deal.
This means that option 1) "more than 12.5 megabytes" is the correct answer.
For any value of x greater than 12.5, the charge for Plan B will be less than the charge for Plan A, making it a better deal for the customer.
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After long study, tree scientists conclude that a eucalyptus tree will
3
grow at the rate of +
ft. per years, where t is time in years. Find the
5 (t+1)³
number of feet the tree will grow in the first year. Be sure to use the proper
units of measure.
After a long study, tree scientists conclude that a eucalyptus tree will grow at the rate of 3ft per year, where t is time in years. So, the tree will grow 5 feet in the first year.
We have to find the number of feet the tree will grow in the first year, given that 5(t + 1)³. The rate of growth of a tree is given as 3ft/year. Therefore, in the first year, the tree will grow 3 feet.
To find the number of feet the tree will grow in the first year, we substitute t = 0 in the given expression.
5(t + 1)³ = 5(0 + 1)³= 5(1)³= 5(1)= 5. Therefore, the tree will grow 5 feet in the first year.
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Please explain how to do this and what the answer is. The answer with best explaination gets Brainliest
Answer:
102
Step-by-step explanation:
we substitute x by 7
7^2+9(7)-10
49+63-10
=102
Find the equations of the asymptotes of the hyperbola defined by the equation shown below. If necessary, round to the nearest tenth. 100pts
The equations of the asymptotes of the hyperbola are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.
To find the equations of the asymptotes of the hyperbola defined by the equation:
[tex]-25x^2 + 81y^2 + 100x + 1134y + 1844 = 0[/tex]
We can rewrite the equation in the standard form by isolating the x and y terms:
[tex]-25x^2 + 100x + 81y^2 + 1134y + 1844 = 0[/tex]
Rearranging the terms:
[tex]-25x^2 + 100x + 81y^2 + 1134y = -1844[/tex]
Next, let's complete the square for both the x and y terms:
[tex]-25(x^2 - 4x) + 81(y^2 + 14y) = -1844\\-25(x^2 - 4x + 4 - 4) + 81(y^2 + 14y + 49 - 49) = -1844\\-25((x - 2)^2 - 4) + 81((y + 7)^2 - 49) = -1844[/tex]
Expanding and simplify
[tex]-25(x - 2)^2 + 100 - 81(y + 7)^2 + 3969 = -1844\\-25(x - 2)^2 - 81(y + 7)^2 = -1844 - 100 - 3969\\-25(x - 2)^2 - 81(y + 7)^2 = -4913[/tex]
Dividing both sides by -4913:
[tex](x - 2)^2/(-4913/25) - (y + 7)^2/(-4913/81) = 1[/tex]
Comparing this equation to the standard form of a hyperbola:
[tex](x - h)^2/a^2 - (y - k)^2/b^2 = 1[/tex]
We can determine that the center of the hyperbola is (h, k) = (2, -7). The value of [tex]a^2[/tex] is (-4913/25), and the value of [tex]b^2[/tex] is (-4913/81).
The equations of the asymptotes can be found using the formula:
y - k = ±(b/a)(x - h)
Substituting the values we found:
y + 7 = ±(√(-4913/81) / √(-4913/25))(x - 2)
Simplifying:
y + 7 = ±(√(4913) / √(81)) × √(25/4913) × (x - 2)
y + 7 = ±(√(4913) / 9) × √(25/4913) × (x - 2)
Rationalizing the denominators and simplifying:
y + 7 = ±(5/9) ×(x - 2)
Finally, rearranging the equation to isolate y:
y = ±(5/9)x - 10/9 - 7
Simplifying further:
y = ±(5/9)x - 79/9
In light of this, the equations for the hyperbola's asymptotes are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.
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Answer:
[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]
Step-by-step explanation:
First, rewrite the given equation in the standard form of a hyperbola by completing the square.
Given equation:
[tex]-25x^2+81y^2+100x+1134y+1844=0[/tex]
Arrange the equation so all the terms with variables are on the left side and the constant is on the right side:
[tex]-25x^2+100x+81y^2+1134y=-1844[/tex]
Factor out the coefficient of the x² term and the coefficient of the y² term:
[tex]-25(x^2-4x)+81(y^2+14y)=-1844[/tex]
Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:
[tex]-25(x^2-4x+4)+81(y^2+14y+49)=-1844-25(4)+81(49)[/tex]
Factor the two perfect trinomials on the left side and simplify the right side:
[tex]-25(x-2)^2+81(y+7)^2=2025[/tex]
Divide both sides by the number of the right side so the right side equals 1:
[tex]\dfrac{-25(x-2)^2}{2025}+\dfrac{81(y+7)^2}{2025}=\dfrac{2025}{2025}[/tex]
[tex]\dfrac{-(x-2)^2}{81}+\dfrac{(y+7)^2}{25}=1[/tex]
[tex]\dfrac{(y+7)^2}{25}-\dfrac{(x-2)^2}{81}=1[/tex]
As the y²-term is positive, the hyperbola is vertical (opening up and down).
The standard equation of a vertical hyperbola is:
[tex]\boxed{\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1}[/tex]
Therefore, comparing this with the rewritten equation:
h = 2k = -7a² = 25 ⇒ a = 5b² = 81 ⇒ b = 9The formula for the equations of the asymptotes of a vertical hyperbola is:
[tex]\boxed{y=\pm \dfrac{a}{b}(x-h)+k}[/tex]
Substitute the values of h, k, a and b into the formula:
[tex]y=\pm \dfrac{5}{9}(x-2)-7[/tex]
Therefore, the equations for the asymptotes are:
[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]
Whats the answer to this?
Answer: [tex]x=200[/tex]
Step-by-step explanation:
[tex]\frac{1}{5}x-\frac{2}{3}y=30[/tex]
[tex]\frac{1}{5}x-\frac{2}{3}(15)=30[/tex]
[tex]\frac{1}{5}x-10=30[/tex]
[tex]\frac{1}{5}x=40[/tex]
[tex]x=\frac{40}{\frac{1}{5}}[/tex]
[tex]x=40*\frac{5}{1}[/tex]
[tex]x=200[/tex]
Given the circle below with tangent NO and
secant QPO. If NO = 18 and Q0 = 27, find
the length of PO. Round to the nearest tenth if necessary.
Answer:
PO = 12
Step-by-step explanation:
given a tangent and a secant from an external point to the circle then
the product of the measures of the secant's external part and the entire secant is equal to the square of the measure of the tangent , that is
OP × OQ = NO²
OP × 27 = 18² = 324 ( divide both sides by 27 )
OP = 12
GEOMETRY 80POINTS
ty
Answer:
37.98
Step-by-step explanation:
IfmWF = 143° and m/WBF = 117°. find mVL
Answer:
arc VL = 91°
Step-by-step explanation:
the chord- chord angle WBF is half the sum of the arcs intersected by the angle and its vertical angle , that is
[tex]\frac{1}{2}[/tex] (WF + VL) = ∠ WBF
[tex]\frac{1}{2}[/tex] (143 + VL ) = 117° ( multiply both sides by 2 to clear the fraction
143° + VL = 234° ( subtract 143° from both sides )
VL = 91°
Problem
Express
0.0939
0.09390, point, 0939 as a fraction.
Answer:
m
Step-by-step explanation:
Find the value of x.
*
20%
x=
104
degrees
Answer:
x=4 explanation: I got it right
Select the correct answer.
The number of hours that 20 people spent watching television per day, in relation to age, is graphed. This quadratic equation represents the model
for the set of data.
y = 0.004z²0.314z + 7.5
Based on the model, approximately how much time does an 18-year-old spend watching television each day?
O A.
OB.
O C.
O D.
3 hours
2 hours
7.5 hours
0.5 hour
Based on the quadratic function, an 18 year old would spend 3 hours watching television.
Using the quadratic function given :
y = 0.004z²-0.314z + 7.5The age is represented as the variable , 'z'
substitute z = 18 into the equation
y = (0.004*18²) - 0.314(18) + 7.5
y = 3.144
y = 3 hours approximately
Hence, an 18 year old spend approximately 18 hours watching television.
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Rotate the triangle 180 counterclockwise around the origin and enter the coordinates. Enter the number that belongs in the green box A (1,-1) B (4,-2) C (2,-4)
Answer:
A''(-1, 1), B''(-4, 2), C''(-2, 4)
Step-by-step explanation:
When rotated 180°, the symbol of the coordinames change
i.e., if x = 2 it becomes x = -2 and if y = -4, it becomes y = 4
so the coordinates A(1, -1), B(4, -2), C(2, -4)
change to A''(-1, 1), B''(-4, 2), C''(-2, 4)
Properties of a determinant
Answer:
Reflection Property.
All-zero Property.
Proportionality.
Switching property.
Factor property.
Scalar multiple properties.
Sum property.
Triangle property.
Bronson Building Inc. is considering a possible investment project, consisting of constructing an office building and then renting it out for use to various local businesses. The initial cost of acquiring the land and constructing the building (first cost) is $21,000,000. The building is expected to be sold for $2,000,000 in 23 years, at the end of the last year of the project. Annual revenue from collecting rents is expected to be $4,000,000, while annual maintenance and operating expenses are projected to equal $2,000,000. Using MARR of 10%, compute the present worth of the project. Note: if the present worth is negative you must include the negative sign with your answer
Answer:The present worth of the project for Bronson Building Inc is $6,389,137.
In order to calculate the present worth, follow these steps:
1. The given information is:
Initial cost (first cost) = $18,000,000
Annual revenue = $5,000,000
Annual expenses = $2,000,000
Net annual cash flow = Annual revenue - Annual expenses = $5,000,000 - $2,000,000 = $3,000,000
MARR = 11%
Project duration = 18 years
Sale price at the end of the project = $8,000,000
2. To calculate the present worth, we first need to find the present value of the net annual cash flows using the MARR as the discount rate. Then, we will add the present value of the sale price and subtract the initial cost.
Present value of net annual cash flows (PV_ACF) = Net annual cash flow * [(1 - (1 + MARR)^(-duration)) / MARR]
PV_ACF = $3,000,000 * [(1 - (1 + 0.11)^(-18)) / 0.11] = $3,000,000 * 7.696 = $23,088,000
3. Find the present value of the sale price at the end of the project.
Present value of sale price (PV_SP) = Sale price / (1 + MARR)^duration
PV_SP = $8,000,000 / (1 + 0.11)^18 = $8,000,000 / 6.146 = $1,301,137
4. Calculate the present worth of the project.
Present worth = PV_ACF + PV_SP - Initial cost
Present worth = $23,088,000 + $1,301,137 - $18,000,000 = $6,389,137
Step-by-step explanation:
advanced functions
solve 4(8-2x)=256
Answer:x=-28
Step-by-step explanation:
Distribute the 4 on the left side of the equation:
32 - 8x = 256
Move the constant term to the right side of the equation:
-8x = 256 - 32
-8x = 224
Divide both sides of the equation by -8 to isolate x:
x = 224 / -8
x = -28
Pamela had $17. She bought 7 burgers for $5.50 and 2 kilograms of orange for $5.30. Find the remaining amount she has now.
$4.20
$5
$6
$6.20
Answer:
$ 6.20 Cents
Step-by-step explanation:
17 - 5.50= 11.5
11.50 - 5.30= 6.2
Add A Zero at the end
You Get 6.20
Movimiento en linea recta
El movimiento en línea recta se refiere al desplazamiento de un objeto en una trayectoria rectilínea, es decir, sin cambios de dirección.
En este tipo de movimiento, la velocidad y la aceleración del objeto pueden variar, pero su dirección se mantiene constante a lo largo del recorrido.
El movimiento en línea recta puede ser uniforme o no uniforme. En el caso del movimiento uniforme, la velocidad del objeto es constante, lo que implica que el desplazamiento realizado en intervalos iguales de tiempo es también constante.
Por otro lado, en el movimiento no uniforme, la velocidad cambia a lo largo del tiempo, resultando en diferentes desplazamientos en intervalos de tiempo iguales.
La descripción matemática del movimiento en línea recta se basa en conceptos como la posición, la velocidad y la aceleración. La posición se refiere a la ubicación del objeto en relación a un punto de referencia, la velocidad representa la tasa de cambio de la posición y la aceleración indica la tasa de cambio de la velocidad.
El estudio del movimiento en línea recta es fundamental en la física y tiene aplicaciones en diversas áreas, como la mecánica, la cinemática, la dinámica y la física de partículas.
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Lashonda is on a game show. She will choose a box to see if she wins a prize. The odds in favor of Lashonda winning a prize are 9/2
. Find the probability of Lashonda winning a prize.
The probability of Lashonda winning a prize is 9/11.
To find the probability of Lashonda winning a prize, we can use the odds given. The odds in favor of Lashonda winning a prize are expressed as 9/2.
Odds are typically represented as a ratio of favorable outcomes to unfavorable outcomes.
In this case, the favorable outcomes are Lashonda winning a prize, and the unfavorable outcomes are Lashonda not winning a prize.
The odds in favor of Lashonda winning a prize can be written as 9:2, where 9 represents the favorable outcomes and 2 represents the unfavorable outcomes.
To calculate the probability, we add the favorable and unfavorable outcomes to get the total number of possible outcomes.
In this case, the total number of outcomes is 9 + 2 = 11.
The probability of Lashonda winning a prize can be calculated as the ratio of the favorable outcomes to the total number of outcomes:
Probability = Favorable outcomes / Total outcomes
Probability = 9 / 11.
Therefore, the probability of Lashonda winning a prize is 9/11.
In conclusion, based on the given odds in favor of Lashonda winning a prize being 9/2, the probability of Lashonda winning a prize is 9/11.
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Question 23 of 41
What is the name of the Platonic solid shown below?
A. Octahedron
B. Dodecahedron
C. Hexahedron
D. Icosahedron
Answer:
That Platonic solid is a dodecahedron.
B is the correct answer.
at the movie theatre, child admission is $5.20 and adult admission is $9.60 on sunday, 131 tickets were sold for a total sales of $1020.00 how many adult tickets were sold that day
Taking into account the definition of a system of linear equations, on Sunday 77 adult tickets were sold.
Definition of System of linear equationsA system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.
Solving a system of equations consists of finding the value of each unknown with which when replacing, they must give the solution proposed in both equations.
Amount of adult tickets soldIn this case, a system of linear equations must be proposed taking into account that:
"a" is the amount of adult tickets sold."c" is the amount of children tickets sold.You know:
At the movie theatre, child admission is $5.20 and adult admission is $9.60 On sunday, 131 tickets were sold for a total sales of $1020.00So, the system of equations to be solved is
a + c= 131
9.60a + 5.20c= 1020
There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.
In this case, isolating the variable c from the first equation:
c= 131 -a
Substituting the expression in the second equation:
9.60a + 5.20×(131 -a)= 1020
Solving:
9.60a + 5.20×131 -5.20a= 1020
9.60a + 681.2 -5.20a= 1020
9.60a -5.20a= 1020 - 681.2
4.4a= 338.8
a= 338.8÷ 4.4
a= 77
In summary, 77 adult tickets were sold that day.
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Find Tan A 6-11, please?
Answer:
5) tan A = 0.42
6) Acute angle is less than 90°
7) Right angle is exactly 90°
8) Obtuse angle is greater than 90° but less than 180°
9) Straight angle is exactly 180°
10) Complementary angles add up to 90°
11) Supplementary angles add up to 180°
Step-by-step explanation:
tan A = opposite / adjacent
= 5/12
= 0.42
The radius of a circle is 3 meters. What is the area of a sector bounded by a 90° arc?
Give the exact answer in simplest form.
Answer:
(90°/360°)π(3^2) = (1/4)(9π) = 9π/4 m²
Use inductive reasoning to predict the most probable next number in the list.
3, 9, -3, 3, -9, -3, -15, -9, -21, ?
Need Help
Answer: -27
Step-by-step explanation: Use inductive reasoning to predict the most probable next number in the list.
3, 9, -3, 3, -9, -3, -15, -9, -21, ?
We can start by looking at the differences between consecutive terms in the list:
9 - 3 = 6 -3 - 9 = -12 3 - (-3) = 6 -9 - 3 = -12 -3 - (-9) = 6 -15 - (-3) = -12 -9 - (-15) = 6 -21 - (-9) = -12
Notice that the differences alternate between positive 6 and negative 12. This suggests that the pattern involves adding 6, then subtracting 12, and then adding 6 again. Applying this pattern to the last term in the list (-21), we get:
-21 + 6 = -15 -15 - 12 = -27 -27 + 6 = -21
Therefore, we predict that the most probable next number in the list is -27.
Para tener una sucesión es imprescindible que los números que lo forman :
A: sean infinitos
B: tengan una ley de formación C:estén ordenados
For a collection of numbers to be considered a sequence, it is essential that they have a law of formation, are ordered in a specific manner, and can be either finite or infinite.
B: They have a law of formation:
A sequence is a set of numbers arranged in a specific order according to a rule or pattern. The numbers in a sequence are not random but follow a specific law of formation.
This law can be a mathematical formula, a recursive relationship, or any other systematic pattern that determines the values of the sequence. Without a well-defined law of formation, a collection of numbers cannot be considered a sequence.
C: They are ordered:
In a sequence, the numbers are arranged in a specific order or sequence. The order of the numbers is crucial and defines the pattern and structure of the sequence.
Each number in the sequence has a unique position or index that determines its place in the sequence. The order of the numbers allows us to identify the next number or predict the pattern of the sequence. Without the concept of order, the numbers would simply be a set of unrelated elements and not a sequence.
A: They may or may not be infinite:
Sequences can be finite or infinite. A finite sequence has a specific number of terms, and once the pattern or rule is established, the sequence ends.
On the other hand, an infinite sequence continues indefinitely, and its terms extend infinitely in one direction or both directions. Whether a sequence is finite or infinite depends on the context and the specific rule or pattern that governs its formation.
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Note: the translated question is:
To have a sequence it is essential that the numbers that form it:
A: be infinite
B: they have a law of formation C: they are ordered
Need help with this question. PLS helpppppp
Answer:
x = 0.39 or
x = -1.72
Step-by-step explanation:
The quadrateic formula is:
[tex]x = \frac{-b\pm\sqrt{b^2 - 4ac} }{2a}[/tex]
eq: 3x² + 4x - 2
which is of the form ax² + bx + c = 0
where a = 3, b = 4 and c = -2
sub in quadratic formuls,
[tex]x = \frac{-4\pm\sqrt{4^2 - 4(3)(-2)} }{2(3)}\\\\=\frac{-4\pm\sqrt{16 + 24} }{6}\\\\=\frac{-4\pm\sqrt{40} }{6}\\\\=\frac{-4\pm2\sqrt{10} }{6}\\\\=\frac{-2\pm\sqrt{10} }{3}\\\\=\frac{-2+\sqrt{10} }{3} \;or\;=\frac{-2-\sqrt{10} }{3}\\\\=0.39 \;or\; -1.72[/tex]
4/5 x 5/7 x 7/9 x 9/11 x 11/13 x 13/15 x 15/17 x 17/19 x 19/21 x 21/23 x 23/25 x 25/27 27/28 x 29/37 x 31/35 x 33/33 x 35/31 x 37/29
The product of the given fractions is 224/61.
To calculate the product of the given fractions, let's simplify and cancel out any common factors.
We have:
(4/5) x (5/7) x (7/9) x (9/11) x (11/13) x (13/15) x (15/17) x (17/19) x (19/21) x (21/23) x (23/25) x (25/27) x (27/28) x (29/37) x (31/35) x (33/33) x (35/31) x (37/29)
Starting from the numerator and denominator of the first fraction, we observe the following cancellations:
5 in the numerator and denominator cancel out.
7 in the numerator and denominator cancel out.
9 in the numerator and denominator cancel out.
11 in the numerator and denominator cancel out.
13 in the numerator and denominator cancel out.
15 in the numerator and denominator cancel out.
17 in the numerator and denominator cancel out.
19 in the numerator and denominator cancel out.
21 in the numerator and denominator cancel out.
23 in the numerator and denominator cancel out.
25 in the numerator and denominator cancel out.
27 in the numerator and denominator cancel out.
Now, let's multiply the remaining fractions:
(4/1) x (1/28) x (29/37) x (31/1) x (1/35) x (37/29)
This simplifies to:
(4 x 1 x 29 x 31 x 37) / (1 x 28 x 37 x 29 x 35)
Simplifying further:
(4 x 29 x 31) / (28 x 35)
We can calculate this to get the final result:
(4 x 29 x 31) / (28 x 35) = 3584 / 980 = 224 / 61.
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What additional information could be used to prove that ΔXYZ ≅ ΔFEG using ASA or AAS?
We would need to demonstrate that two angles of one triangle are congruent to two angles of the other triangle, and a pair of non-included sides are congruent.
To prove that triangles ΔXYZ and ΔFEG are congruent using the ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side) congruence criteria, we need to show that they share certain corresponding angles and sides.
In ASA, we would need to show that both triangles have two congruent angles and the included side between those angles is congruent. In AAS, we would need to demonstrate that two angles of one triangle are congruent to two angles of the other triangle, and a pair of non-included sides are congruent.
Additional information that could be used to prove the congruence of ΔXYZ and ΔFEG using ASA or AAS includes:
1. Angle X = Angle F: If we can show that angle X in triangle ΔXYZ is congruent to angle F in triangle ΔFEG, we have one angle congruence.
2. Angle Y = Angle E: If we can demonstrate that angle Y in triangle ΔXYZ is congruent to angle E in triangle ΔFEG, we have the second angle congruence.
3. Side XY = Side FE: If we can prove that side XY in triangle ΔXYZ is congruent to side FE in triangle ΔFEG, we have the included side congruence.
Alternatively:
4. Angle Z = Angle G: If we can show that angle Z in triangle ΔXYZ is congruent to angle G in triangle ΔFEG, we have the second angle congruence in AAS.
5. Angle Y = Angle E: If we can demonstrate that angle Y in triangle ΔXYZ is congruent to angle E in triangle ΔFEG, we have the second angle congruence in AAS.
6. Side XZ = Side FG: If we can prove that side XZ in triangle ΔXYZ is congruent to side FG in triangle ΔFEG, we have a pair of non-included side congruence.
By establishing these angle and side congruences, we can use either the ASA or AAS congruence criteria to prove that triangles ΔXYZ and ΔFEG are congruent.
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Mason plans to study for 1 and 1-half hours. Once he has studied for 1-third of the planned time, he will take a break. Mason has been studying for 12 minutes.
Question
How many ,begin emphasis,more,end emphasis, minutes does Mason need to study before he takes a break? Enter the answer in the box.
Response area with 1 text input box
Answer:
He needs to study for an additional 30 minutes - 12 minutes = 18 minutes before taking a break.
Step-by-step explanation:
To determine how many more minutes Mason needs to study before taking a break, we can calculate the remaining study time.
Mason plans to study for 1 and 1-half hours, which is equivalent to 90 minutes.
He will take a break once he has studied for 1-third of the planned time, which is 1/3 * 90 minutes = 30 minutes.
Mason has already studied for 12 minutes.
Therefore, he needs to study for an additional 30 minutes - 12 minutes = 18 minutes before taking a break.
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Net Present Value Method, Internal Rate of Return Method, and Analysis
The management of Advanced Alternative Power Inc. is considering two capital investment projects. The estimated net cash flows from each project are as follows:
Year Wind Turbines Biofuel Equipment
1 $420,000 $880,000
2 420,000 880,000
3 420,000 880,000
4 420,000 880,000
Present Value of an Annuity of $1 at Compound Interest
Year 6% 10% 12% 15% 20%
1 0.943 0.909 0.893 0.870 0.833
2 1.833 1.736 1.690 1.626 1.528
3 2.673 2.487 2.402 2.283 2.106
4 3.465 3.170 3.037 2.855 2.589
5 4.212 3.791 3.605 3.352 2.991
6 4.917 4.355 4.111 3.784 3.326
7 5.582 4.868 4.564 4.160 3.605
8 6.210 5.335 4.968 4.487 3.837
9 6.802 5.759 5.328 4.772 4.031
10 7.360 6.145 5.650 5.019 4.192
The wind turbines require an investment of $1,199,100, while the biofuel equipment requires an investment of $2,278,320. No residual value is expected from either project.
Required:
1a. Compute the net present value for each project. Use a rate of 10% and the present value of an annuity of $1 in the table above. If required, use the minus sign to indicate a negative net present value. If required, round to the nearest whole dollar.
Wind Turbines Biofuel Equipment
Present value of annual net cash flows $fill in the blank 1 $fill in the blank 2
Less amount to be invested $fill in the blank 3 $fill in the blank 4
Net present value $fill in the blank 5 $fill in the blank 6
1b. Compute a present value index for each project. If required, round your answers to two decimal places.
Present Value Index
Wind Turbines fill in the blank 7
Biofuel Equipment fill in the blank 8
2. Determine the internal rate of return for each project by (a) computing a present value factor for an annuity of $1 and (b) using the present value of an annuity of $1 in the table above. If required, round your present value factor answers to three decimal places and internal rate of return to the nearest whole percent.
Wind Turbines Biofuel Equipment
Present value factor for an annuity of $1 fill in the blank 9 fill in the blank 10
Internal rate of return fill in the blank 11 % fill in the blank 12 %
3. The net present value, present value index, and internal rate of return all indicate that the
is a better financial opportunity compared to the
, although both investments meet the minimum return criterion of 10%.
1a. Compute NPV by calculating the present value of net cash flows and subtracting the investment amount.
1b. Compute PVI by dividing NPV by the investment amount.
2. Determine IRR by finding the discount rate corresponding to an NPV of zero.
3. Compare NPV, PVI, and IRR to identify the better financial opportunity.
1a. To compute the net present value (NPV) for each project, we need to calculate the present value of the annual net cash flows and subtract the amount to be invested. Using the present value of an annuity of $1 from the table, we can fill in the following values:
Wind Turbines:
Present value of annual net cash flows: $420,000 * 1.736 + $420,000 * 2.487 + $420,000 * 3.170 + $420,000 * 3.791
Less amount to be invested: $1,199,100
Net present value: NPV_Wind_Turbines = Present value of annual net cash flows - Amount to be invested
Biofuel Equipment:
Present value of annual net cash flows: $880,000 * 1.736 + $880,000 * 2.487 + $880,000 * 3.170 + $880,000 * 3.791
Less amount to be invested: $2,278,320
Net present value: NPV_Biofuel_Equipment = Present value of annual net cash flows - Amount to be invested
1b. The present value index (PVI) can be calculated by dividing the NPV by the amount to be invested:
Present Value Index = NPV / Amount to be invested
2. To determine the internal rate of return (IRR) for each project, we need to find the discount rate at which the NPV becomes zero. We can use the present value of an annuity of $1 from the table to calculate the present value factor for an annuity of $1. Then, we can find the discount rate that corresponds to an NPV of zero.
Wind Turbines:
Present value factor for an annuity of $1: Fill in the values from the table
Internal rate of return: IRR_Wind_Turbines = Discount rate corresponding to NPV = 0
Biofuel Equipment:
Present value factor for an annuity of $1: Fill in the values from the table
Internal rate of return: IRR_Biofuel_Equipment = Discount rate corresponding to NPV = 0
3. Based on the calculations of NPV, PVI, and IRR, we can compare the two projects. The project with the higher NPV, PVI, and IRR is considered the better financial opportunity. Both investments meet the minimum return criterion of 10%, but the project with the higher financial indicators is preferred.
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