The container that holds the water for the football team is 2/5 full. After pouring in 6 gallons of water, it’s 7/10 full. How many gallons can the container hold

Here we go ~

[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

[tex]\qquad \sf  \dashrightarrow \: (1 - \tan {}^{4} (a) ) \cos {}^{4} (a) [/tex]

[tex]\qquad \sf  \dashrightarrow \: (1 + \tan {}^{2} (a) )(1 - { \tan}^{2} (a)) \cos {}^{4}(a ) [/tex]

[ a² - b² = (a + b)(a - b) ]

[tex]\qquad \sf  \dashrightarrow \: ( \sec {}^{2} (a) )(1 - ( \sec{}^{2} (a) - 1) )\cos {}^{4} (a) [/tex]

[ sec² a = 1 + tan² a, so : tan² a = sec²a - 1 ]

[tex]\qquad \sf  \dashrightarrow \: \bigg( \dfrac{1}{{}cos^{2} (a)} \bigg)(2 - \sec{}^{2} (a) ) \cos {}^{4} (a) [/tex]

[tex]\qquad \sf  \dashrightarrow \: \bigg(2 - \dfrac{1}{ \cos {}^{2} (a) } \bigg) \cos {}^{2} (a) [/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{2 \cos {}^{2} (a) - 1}{ \cos {}^{2} (a) } \times \cos {}^{2} (a) [/tex]

[tex]\qquad \sf  \dashrightarrow \: 2 \cos {}^{2} (a) - 1[/tex]

[tex]\qquad \sf  \dashrightarrow \: 2(1 - \sin {}^{2} (a) ) - 1[/tex]

[ sin²a + cos² a = 1, hence sin²a = 1 - cos²a ]

[tex]\qquad \sf  \dashrightarrow \: 2 - 2 \sin {}^{2} (a) - 1[/tex]

[tex]\qquad \sf  \dashrightarrow \: 1 - 2 \sin {}^{2} (a) [/tex]

Answer:

See proof below

Step-by-step explanation:

Prove [tex]\left(\:1\:-\:tan^4A\right)cos^4A\:=\:1\:-\:2\:sin^2A[/tex]

[tex]\left(1-\tan ^4\left(A\right)\right)\cos ^4\left(A\right)[/tex]  can be expressed in sin, cos terms

Use the trigonometric identity [tex]\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}[/tex]

[tex]\left(1-\tan ^4\left(A\right)\right)\cos ^4\left(A\right) = \left(1-\left(\frac{\sin \left(A\right)}{\cos \left(A\right)}\right)^4\right)\cos ^4\left(A\right)[/tex]

[tex]\mathrm{Simplify}\:\left(1-\left(\frac{\sin \left(A\right)}{\cos \left(A\right)}\right)^4\right)\cos ^4\left(A\right)[/tex]

[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}[/tex]

= [tex]\left(1-\frac{\sin ^4\left(A\right)}{\cos ^4\left(A\right)}\right)\cos ^4\left(A\right)[/tex]

Multiplying the expression in parentheses by [tex]\cos ^4\left(A\right)[/tex] we get

[tex]\frac{\cos ^4\left(A\right)-\sin ^4\left(A\right)}{\cos ^4\left(A\right)}\cos ^4\left(A\right)[/tex]

Cancel the common factor [tex]\cos ^4\left(A\right)[/tex]

This gives us

[tex]\cos ^4\left(A\right)-\sin ^4\left(A\right)[/tex]

Now,

[tex]\sin ^4\left(A\right)=\left(\sin ^2\left(A\right)\right)^2[/tex]

[tex]\cos ^4\left(A\right)=\left(\cos ^2\left(A\right)\right)^2[/tex]

[tex]\:\cos ^4\left(A\right)-\sin ^4\left(A\right)[/tex] = [tex]\left(\cos ^2\left(A\right)\right)^2-\left(\sin ^2\left(A\right)\right)^2[/tex]

[tex]\mathrm{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}x^2-y^2=\left(x+y\right)\left(x-y\right)[/tex]

[tex]\left(\cos ^2\left(A\right)\right)^2-\left(\sin ^2\left(A\right)\right)^2=\left(\cos ^2\left(A\right)+\sin ^2\left(A\right)\right)\left(\cos ^2\left(A\right)-\sin ^2\left(A\right)\right)[/tex]

= [tex]\cos ^2\left(A\right)-\sin ^2\left(A\right)[/tex]   [tex]\textrm{ since }\cos ^2\left(A\right)+\sin ^2\left(A\right)[/tex] = 1

Using the fact that [tex]\cos ^2\left(A\right)=1-\sin ^2\left(A\right)[/tex]

we get

[tex]\cos ^2\left(A\right)-\sin ^2\left(A\right) = 1-\sin ^2\left(A\right)-\sin ^2\left(A\right)\\\\= 1-2\sin ^2\left(A\right)[/tex]

Proved

Answer:

(x,y) = (-1,1)

Step-by-step explanation:

step 1

subsitute the given value of y into the equation

2x+7(x+2)=5

step 2

solve for x

x= -1

step 3

subsitute the value of x in to the second equation

y= -1+2

step 4

solve for y

y=1

step 5

check if there true by putting the value of x and y back in to the equation.

then configure x and y into an ordered pair.

(x,y)=(-1,1)

I hope this helped!

Answer:

D

Step-by-step explanation:

a linear expression in two variables has the form ax + by

an inequality uses the symbols < , > , ≤ , ≥

the only expression in this form is

3x + 4y ≥ 10

Answer:

A. The expression is in its simplest form.

Step-by-step explanation:

Answer:

i tried this i think the answer is 11a plus 2a subract3b ppus 5

Answer:x=

Step-by-step explanation:

the standard form equation is y=ax²+bx+c where a is the quadratic coefficient, b is the linear coefficient, and c is the constant coefficient.

therefore, for y =  5/16x² - 5x/4 -35/4

Quadratic coefficient = 5/16, linear coefficient = 5/4, constant term = -35/4

What are zeros ?

zeros denotes the factors of the given equation in other words the zeros of the function are the values that make the factors zero. The factors need to multiply out to give the original standard-form equation.

Polynomial roots are the same as polynomial zeros, so they can be found by factoring the quadratic equation into two linear factors, after which they can be equating to zero.

Its easiest to first start out with a vertex form equation because it can then be converted to a standard quadratic equation.

given a vertex at (2,-10) , and point of intersection at (6,-5), the equation can be set up like this in the form of :

y = a(x-h)²+ k

y = a(x-2)^2-10, as we know h and k from the vertex.

We also know y and x from the given point of intersection so a can be solved by substituting the values of x and y to get value of a.

y = a(x-2)²-10

-5 = a(6-2)² - 10

16a = 10-5

a = 5/16

a = 5/16 which is also known as the quadratic coefficient because it is part of a second degree quantity and a Trinomial as a whole(quadratic).

since all the variables are known, you can expand the equation and set it to standard form :

y = 5/16(x-2)²-10

y = 5/16(x-2)² - 10

y = 5/16(x²+4-4x) -10

y = 5/16x² + 5/4 - 5x/4 - 10

y =  5/16x² - 5x/4 -35/4

For reference, the standard form equation is y=ax²+bx+c where a is the quadratic coefficient, b is the linear coefficient, and c is the constant coefficient.

In this instance, 5/16x² - 5x/4 -35/4 = ax²+bx+c

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Answer: 8 and 9

Step-by-step explanation:

square root of - 80 is 8.94427191 so 8 and 9 is the integers around it.

Answer:

200mm

Step-by-step explanation:

200mm can be rewritten as 20 cm

a womans hand is on average 17.3 cm

this is the closest estimate out of the three

Answer: 29/21

Quotient: 1

Reminder: 8

Step-by-step explanation:

Add up the amount of 7th graders playing instruments (not including drums) total and put that number under the amount of 8th graders playing a instrument.

Use the 1st  digit 2 from dividend 29

21 & 29 squared root

Since 2 is less than 21, use the next digit 9 from dividend 29 and add 0 to the quotient. Find the closest multiple of 21 to 29.

1×21=21 is the nearest. Now subtract 21 from 29 to get reminder 8. Add 1 to quotient.

Since 8 is less than 21, stop the division. The reminder is 8. The topmost line 01 is the quotient. Remove all zeros at the start of the quotient to get the actual quotient 1.

The expression sec x csc x cot x  in terms of sin x and cos x is

1/ (sin x)^2.

According to the given question.

We have an expression sec x csc x cot x.

Here, we have to write the above expression in terms of sinx and cosx.

As we know that,

sec x = 1/cos x

csc x = 1/sin x

cot x = cos x/ sin x

Thereofore, the expression sec x csc x cot x  in terms of sin x and cos x is given by

sec x csc x cot x

= (1/cos x) (1/sinx) (cosx/ sinx)

= 1/ (sin x)^2

Hence, the expression sec x csc x cot x  in terms of sin x and cos x is

1/ (sin x)^2.

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The elements that will be coming in the set D∪E will be {12, 14, 15, 16, 17}.

A set may be defined as a collection of letters or numbers that are written in order to depict a certain value or entity. A set is always represented by a capital letter symbol and is always written in curly brackets { }. Union of two sets may be defined as a new set which has the collection of all the elements of the two individual sets. Union of two sets is represented by the symbol '∪'. Union of two Indvidual sets, set A and set b is written as A∪B.

Now, according to the question set D is = {12, 15, 17} and set E is = {12, 14, 15, 16}.

The union of the two sets contains all the elements of the sets D and E.

Thus, D∪E will be given by {12, 14, 15, 16, 17}.

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Coordinates of P is 8

Since 3+2 = 5, we are dividing line segment AB into 5 congruent parts with point P 3/5 of the way from A to B.  

x-coordinate of P = x-coordinate of A + (3/5)(change in x from A to B) = -4 + (3/5)(20) = 8

y-coordinate of P = y-coordinate of A + (3/5)(change in y from A to B) = 8 + (3/5)(-10) = 2

P = (8,2)

In each axis, find the distance from A to B, apply the ratio to get the distance from A to P, and add that value to A.

Note the ratio:   We are given the ratio of AP to PB as 3/2.  This means that the ratio of AP to AB is 3/5

Here is is for the X-axis:

AB = 20

AP = 20 *3/5 = 12

So P = -4 + 12 = 8

It is one of the branches of geometry where the position of a point is defined using coordinates.

Coordinates are a set of values which helps to show the exact position of a point in the coordinate plane.

A coordinate plane is a 2D plane which is formed by the intersection of two perpendicular lines known as the x-axis and y-axis.

It is used to divide any line into two parts, in m:n ratio

The number line which is also known as a Cartesian plane is divided into four quadrants by two axes perpendicular to each other, labelled as the x-axis (horizontal line) and the y-axis(vertical line).

The four quadrants along with their respective values are represented in the graph below-

Quadrant 1 : (+x, +y)

Quadrant 2 : (-x, +y)

Quadrant 3 : (-x, -y)

Quadrant 4 : (+x, -y)

The point at which the axes intersect is known as the origin. The location of any point on a plane is expressed by a pair of values (x, y) and these pairs are known as the coordinates.

How To Find Coordinates of a Point on Graph With Examples

A set of values that shows the exact position of a point in a two-dimensional coordinate plane are called the coordinates. These represent the exact location of a point on a coordinate graph having both x, y axes. You can check the definition of coordinates, step by step detailed procedure to find the coordinates of a point with solved examples.

Coordinates Definition

A pair of numbers that describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines called the coordinates. Usually represented by (x, y) the x value and y value of the point on a graph. Every point or an ordered pair contains two coordinates. The first one is the x coordinate or abscissa and the second is the y coordinate or ordinate. The values of the coordinates of a point can be any positive or negative real number.

The other types of coordinates are map coordinates (north/south, east/west), three-dimensional coordinates, polar coordinates (distance, angle), etc. Detailed information about x coordinate, y coordinate of a point follows:

x‐coordinate (Abscissa): The first number or the number which is located to the left side of a comma in the point is the x coordinate of the ordered pair. It represents the amount of movement along the x-axis from the origin. The movement is to the right side if the number is positive and to the left side of the origin if the number is negative.

y‐coordinate (Ordinate): The number which is located to the right side of the comma in the ordered pair or the second number is known as the y coordinate of the ordered pair. This ordinate indicates the amount of movement along the y-axis. If the number is positive, then the movement is above the origin and the movement is below the origin if the number is negative.

Point on x-axis: A point on the x-axis means its movement along the horizontal line is always zero and the y-coordinate of all points on the x-axis is zero. Therefore, the coordinates of a point on the x-axis are of the form (x, 0).

Point on y-axis: A point on the y-axis means the distance from the y-axis is zero and the x coordinate of every point on the y-axis is zero. Hence, the coordinates of a point on the y-axis are (0, y).

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Answer:

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

using the rule of exponents

[tex]a^{m}[/tex] ÷ [tex]a^{n}[/tex] = [tex]a^{(m-n)}[/tex] , then

[tex]6^{9}[/tex] ÷ 6³ = [tex]6^{(9-3)}[/tex] = [tex]6^{6}[/tex]

Answer:

Minimum = (-3, -2)

Step-by-step explanation:

Standard form of a quadratic function:

[tex]f(x)=ax^2+bx+c[/tex]

If a > 0 the parabola opens upwards and the curve has a minimum point.

If a < 0 the parabola opens downwards and curve has a maximum point.

Given function:

[tex]f(x)=x^2+6x+7[/tex]

As a > 0, the parabola opens upwards and so the curve has a minimum point.

The minimum/maximum point of a quadratic function is its vertex.

Vertex form of a quadratic function:

[tex]f(x)=(x-h)^2+k[/tex]

Where (h, k) is the vertex.

To rewrite the given function in vertex form, complete the square.

Add and subtract the square of half the coefficient of the term in x:

[tex]\implies f(x)=x^2+6x+7 +\left(\dfrac{6}{2}\right)^2-\left(\dfrac{6}{2}\right)^2[/tex]

[tex]\implies f(x)=x^2+6x+7 +9-9[/tex]

[tex]\implies f(x)=x^2+6x+9+(7 -9)[/tex]

[tex]\implies f(x)=x^2+6x+9-2[/tex]

Factor the perfect square trinomial formed by x²+6x+9:

[tex]\implies f(x)=(x+3)^2-2[/tex]

Compare with the vertex form:

Therefore, the vertex is (-3, -2) and so the minimum value of the given function is (-3, -2).

The layout of the fast-food restaurant of dimensions 6 by 8 5 feet squares is presented in the attached table created with Sheets.

The dimensions of the facility are;

Horizontal = 6 squares

Vertical = 8 squares

The side length of each square = 5 feet

Therefore;

Area of each square = 5² ft.² = 25 ft.²

Number of squares, n, required by each dependent are therefore;

CB department, n = 300 ÷ 25 = 12 squares

CF department, n = 200 ÷ 25 = 8 squares

PS department, n = 8 squares

DD department, n = 8 squares

CS department, n = 12 squares

A layout for the fast-food restaurant is therefore;

Please see the attached table layout created using Sheets.

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Answer:

5<x<7

Step-by-step explanation:

Hey there !

firstly, in inequality; we must know that we should put the x alone.

so, in order to put the x alone; we should do the following process,

1</2x-9/<5

here,

adding +9 in all sides, we get;

=1+9<2x-9+9<5+9

=10<2x<14

dividing it by 2;we get

=10/2<2x/2<14/2

=5<x<7

This is the following answer. hope it helps!

Thank you!

Answer:

[tex]\boxed{\sf y^{15}}[/tex]

Step-by-step explanation:

Given expression:

When any expression is given in the format [tex]\sf (\ a^b \ )^c[/tex] then [tex]\sf a^{bc}[/tex].

simplify

Answer:

(-6, 7)

Step-by-step explanation:

90 degrees counterclockwise / 270 degrees clockwise:

(x,y) -> (-y,x)
180 degrees counterclockwise / 180 degrees clockwise:

(x,y) -> (-x,-y)

270 degrees counterclockwise / 90 degrees clockwise:

(x,y) -> (y,-x)

Answer:

2 on the 1st, 1 on the second, 3 on the third, 1 on the fourth

Answer: 31 Minutes

Step-by-step explanation:

Minus 0.80 from 7.93, and then divide 7.13 by 0.23 and you get 31

The two true statements are:

b) (f(5)−f(1)) / (5−1) < 0

d)  f(5) < f(1)

We know that f(x) is a decreasing function, this means that, as x increases, the value of f(x) decreases.

So, if there are two values in the domain a and b, such that a < b, we will have that:

f(a) > f(b).

Then the statement:

[ f(5) - f(1)]/(5 - 1) < 0

Is true, because f(1) is larger than f(5).

Which means that statement d ( f(5) < f(1)) is also true.

Then the two correct options are b and d.

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The total change in the tons of recyclable material taken to the landfill after 10 7 months resulting from the awareness program = (x - 95.9) tons, where x is the initial amount.

In this question, we have been given after a recycling awareness program, the number of tons of recyclable material taken to the landfill is reduced by 13 tons per month.

We need to represent the total change in the tons of recyclable material taken to the landfill after 10 7 months resulting from the awareness program.

Given that reduction in materials = 13 7/10 tons per month

We rewrite the improper fraction 13 7/10 as a proper fraction.

13 7/10 = 137/10

Monthly reduction = 137/10 tons

After 7 months : (7 × 137/10)

                         = 959/10 tons

                         = 95.9 tons

Change in amount of material taken to the land fill:

Initial amount - amount after 7 months

If initial amount be x, then the change in amount = (x - 95.9) tons

So, we get an expression (x - 95.9)

Therefore, the total change in the tons of recyclable material taken to the landfill after 10 7 months resulting from the awareness program =                 (x - 95.9) tons, where x is the initial amount.

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Answer:From the given table, we have that:

1. The relation of the situation is: (20,2), (22.5, 5), (25,4), (27.5, 4), (29,4), (29,5), (30,1), (30,2), (30,3), (30,4), (30,5), (32.5, 1), (32.5, 3), (35,4), (40,2), (42.5, 3).

2. The domain is {20, 22.5, 25, 27.5, 29, 30, 32.5, 35, 40, 42.5} and the range is {1, 2, 3, 4, 5}.

3. There are inputs such as 25, 29, 30 and 32.5 that are mapped to multiple outputs, hence the relation does not represent a function.

What is the relation of this situation?

To build the relation, we look at the graph, and write all pairs of (Age, Finishing Place), as follows:

(20,2), (22.5, 5), (25,4), (27.5, 4), (29,4), (29,5), (30,1), (30,2), (30,3), (30,4), (30,5), (32.5, 1), (32.5, 3), (35,4), (40,2), (42.5, 3).

What are the domain and the range?

The domain of a relation is the set that contains all possible input values for the values, that is, the values of x.

The range of a relation is the set that contains all possible output values for the values, that is, the values of y.

Hence, for the given relation:

The domain is {20, 22.5, 25, 27.5, 29, 30, 32.5, 35, 40, 42.5} and the range is {1, 2, 3, 4, 5}.

When does a relation represents a function?

A relation represents a function if each value of the input is mapped to only one value of the output.

For this problem, we have that:

There are inputs such as 25, 29, 30 and 32.5 that are mapped to multiple outputs, hence the relation does not represent a function.

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Step-by-step explanation:

Answer: $2.50

Step-by-step explanation: To find the tax rate, you need to divide 1.50 by 30

1.5/30 = .05, which is the percent taxed. To find the tax amount for a $50 item, you just need to multiply $50 by the tax rate, which is .05, to get $2.50

Answer:

72

Step-by-step explanation:

let's begin by restating what we already know,

360 bottles of juice in 5 minutes

for goodness's sake let's abbreviate this

360 boj in 5 mins

RATIO TIME!

360 : 5

pay attention how I put the BOJ (bottles of juice) first and the time last

now to find how many boj per min. we will say

[tex]\frac{1}{5} (360: 5)[/tex]

72:1

so, we now know that the factory of juice produces 72 bottles of juices per minute ( that's ALOT of juices per minute imo)

^_^ hope this helps, happy learning

Answer:

16

Step-by-step explanation: This is because when we look at the square we realize by definition it is a 4-sided shape with 4 equal side lengths this means that if one side is 4 all the 4 sides of a square are also 4 and in this case, we would do length times width and get an answer of an area of 16 units.