What is the meaning of "it will be tacitly understood that each such formula can be written in a form that only involves ∈ and = as nonlogical symbols"?

1. The total in mass of the ingredients used is 2.35kg

2. The cost of sugar needed is R25.78.

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

These statements are interpreted into mathematical equation or expression.

1. The recipes are ;

0.8kg = 800g

sugar = 650g

butter = 900000 mg = 900000/1000 = 900g

Therefore the total mass of ingredients

= 800 + 650 +900

= 2350g

in kilograms, 1000g is 1kg

2350g = 2350/1000

= 2.35kg

2. If 150g = R5.95

1g = 5.95/150

650g = 5.95 × 650/150

= R25.78.

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Answer: To calculate the distance the beetle walks diagonally across the table, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the two sides of the right triangle formed by the table are 160 cm and 90 cm. Let's call the hypotenuse (the distance the beetle walks) "d."

So, applying the Pythagorean theorem, we have:

d^2 = 160^2 + 90^2

Simplifying:

d^2 = 25600 + 8100

d^2 = 33700

Taking the square root of both sides:

d ≈ √33700

d ≈ 183.54 cm

Therefore, the beetle walks approximately 183.54 cm diagonally across the kitchen table.

Answer: is X = 5 within the diagram.

Answer: its D! not 8!!!

Step-by-step explanation:

The value of AB is,

⇒ AB = 5.9

(rounded to nearest tenth)

We have to given that,

A right triangle ABC is shown.

Now, By trigonometry formula,

we get;

⇒ cos 33° = Base / Hypotenuse

Substitute all the values, we get;

⇒ cos 33° = AB / 7

⇒ 0.84 = AB / 7

⇒ AB = 0.84 × 7

⇒ AB = 5.88

⇒ AB = 5.9

(rounded to nearest tenth)

Thus, We get;

AB = 5.9

(rounded to nearest tenth)

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Answer:

32 inches Aprox

Step-by-step explanation:

To find the size of a rectangular television screen given its height and width, we can use the Pythagorean theorem. The diagonal measurement (size) is the hypotenuse of a right triangle formed by the height and width.

Given:

Height of the television screen (h) = 20 inches

Width of the television screen (w) = 25 inches

Using the Pythagorean theorem:

diagonal² = height² + width²

diagonal² = 20² + 25²

diagonal² = 400 + 625

diagonal² = 1025

Taking the square root of both sides:

diagonal ≈ √1025

diagonal ≈ 32.02

Rounding to the nearest inch, the size of the television screen is approximately 32 inches.

Therefore, the size of the rectangular television screen, rounded to the nearest inch, is 32 inches.

Answer:

[tex]\textsf{a)} \quad x = 17[/tex]

[tex]\textsf{b)} \quad T_n=3n^2-3n-1[/tex]

[tex]\textsf{a)} \quad x = 20[/tex]

[tex]\textsf{b)} \quad T_n=3n^2-4n+5[/tex]

Step-by-step explanation:

Given quadratic number pattern:

To find the equation for the nth term, we can use the general form of a quadratic equation:

[tex]\boxed{T_n=an^2 + bn + c}[/tex]

where n is the position of the term.

Let's substitute the values of T₁, T₂ and T₄ into the quadratic equation: to create three equations:

[tex]\begin{aligned}T_1=a(1)^2+b(1)+c&=-1\\a+b+c&=-1\end{aligned}[/tex]

[tex]\begin{aligned}T_2=a(2)^2+b(2)+c&=5\\4a+2b+c&=5\end{aligned}[/tex]

[tex]\begin{aligned}T_4=a(4)^2+b(4)+c&=35\\16a+4b+c&=35\end{aligned}[/tex]

Rearrange the first equation to isolate c:

[tex]c=-a-b-1[/tex]

Substitute this into the second and third equations:

[tex]\begin{aligned}4a+2b+(-a-b-1)&=5\\3a+b&=6\end{ailgned}[/tex]

[tex]\begin{aligned}16a+4b+(-a-b-1)&=35\\15a+3b&=36\end{ailgned}[/tex]

Solve the equations simultaneously by rearranged the first equation to isolate b and substituting this into the second equation and solving for a:

[tex]b=-3a+6[/tex]

[tex]\begin{aligned}15a+3(-3a+6)&=36 \\15a-9a+18&=36\\6a&=18\\a&=3 \end{aligned}[/tex]

Substitute the found value of a into the equation for b and solve for b:

[tex]\begin{aligned}b&=-3a+6\\&=-3(3)+6\\&=-9+6\\&=-3\end{aligned}[/tex]

Finally, substitute the found values of a and b into the equation for c and solve for c:

[tex]\begin{aligned}c&=-a-b-1\\&=-3-(-3)-1\\&=-3+3-1\\&=-1\end{aligned}[/tex]

Therefore, the equation for the nth term is:

[tex]\boxed{T_n=3n^2-3n-1}[/tex]

The value of x is the 3rd term. Therefore, to find the value of x, substitute n = 3 into the equation for the nth term:

[tex]\begin{aligned}T_3&=3(3)^2-3(3)-1\\&=3(9)-3(3)-1\\&=27-9-1\\&=18-1\\&=17\end{aligned}[/tex]

Therefore, the value of x is 17.

[tex]\hrulefill[/tex]

Given quadratic number pattern:

To find the equation for the nth term, we can use the general form of a quadratic equation:

[tex]\boxed{T_n=an^2 + bn + c}[/tex]

where n is the position of the term.

Let's substitute the values of T₁, T₂ and T₄ into the quadratic equation: to create three equations:

[tex]\begin{aligned}T_1=a(1)^2+b(1)+c&=4\\a+b+c&=4\end{aligned}[/tex]

[tex]\begin{aligned}T_2=a(2)^2+b(2)+c&=9\\4a+2b+c&=9\end{aligned}[/tex]

[tex]\begin{aligned}T_4=a(4)^2+b(4)+c&=37\\16a+4b+c&=37\end{aligned}[/tex]

Rearrange the first equation to isolate c:

[tex]c=-a-b+4[/tex]

Substitute this into the second and third equations:

[tex]\begin{aligned}4a+2b+(-a-b+4)&=9\\3a+b&=5\end{ailgned}[/tex]

[tex]\begin{aligned}16a+4b+(-a-b+4)&=37\\15a+3b&=33\end{ailgned}[/tex]

Solve the equations simultaneously by rearranged the first equation to isolate b and substituting this into the second equation and solving for a:

[tex]b=-3a+5[/tex]

[tex]\begin{aligned}15a+3(-3a+5)&=33 \\15a-9a+15&=33\\6a&=18\\a&=3 \end{aligned}[/tex]

Substitute the found value of a into the equation for b and solve for b:

[tex]\begin{aligned}b&=-3a+5\\&=-3(3)+5\\&=-9+5\\&=-4\end{aligned}[/tex]

Finally, substitute the found values of a and b into the equation for c and solve for c:

[tex]\begin{aligned}c&=-a-b+4\\&=-3-(-4)+4\\&=-3+4+4\\&=5\end{aligned}[/tex]

Therefore, the equation for the nth term is:

[tex]\boxed{T_n=3n^2-4n+5}[/tex]

The value of x is the 3rd term. Therefore, to find the value of x, substitute n = 3 into the equation for the nth term:

[tex]\begin{aligned}T_3&=3(3)^2-4(3)+5\\&=3(9)-4(3)+5\\&=27-12+5\\&=15+5\\&=20\end{aligned}[/tex]

Therefore, the value of x is 20.

The height of the average fourth grader is 137cm 1mm. This height can be determined by multiplying 3 by the average height of a newborn baby.

Given information,

The average height of babies = 45 cm 7mm

45 cm 7 mm is equivalent to 45.7 cm (since there are 10 millimeters in a centimeter).

Let the height of a fourth grader be x.

According to the question,

The height of a fourth grader (x) = 3 × the average height of a newborn baby

The height of a fourth grader (x) = 3 × 45.7

The height of a fourth grader (x)= 137.1 = 137cm 1mm

Therefore, the height of a fourth grader is 137cm 1mm.

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Hello!

A = 115° => opposite

B = 115° => corresponding angles

C = 65° => 180° - 115°

Answer:

A = 115 degrees

B = 115 degrees

C = 65 degrees

Step-by-step explanation:

A is opposite of 115 so it is 115 degrees.

B is 115 because it is on a parellel line to the one of 115

C is 180 minus B, which is 180 - 115 = 65

Answer:

Infinite solutions (D).

Step-by-step explanation:

Here is how:

To determine the true statement about the given equation, let's simplify it step by step:

-9(x + 3) + 12 = -3(2x + 5) - 3x

Distributing the -9 and -3 on the left and right sides respectively:

-9x - 27 + 12 = -6x - 15 - 3x

Combining like terms:

-9x - 15 = -9x - 15

Now, let's analyze the equation. We have -9x on both sides, and -15 on both sides. By subtracting -9x from both sides and -15 from both sides, we obtain:

0 = 0

This equation is true regardless of the value of x. In other words, it holds for all values of x. Therefore, the equation has infinitely many solutions.

Answer:

The correct answer is: "The equation has one solution, x = 0"

All the solutions are,

g⁻¹ = { (7, - 7), (- 7, - 6), (6, - 1), (9, 3) }

h⁻¹ (x) = 5x - 13

⇒ (h⁻¹ о h ) (- 3) = 3

We have to given that,

Functions g and f are defined as,

g = {(- 7, 7), (- 6, - 7), (- 1, 6), (3, 9)

And, h (x) = (x + 13) / 5

Now, We can simplify for inverse function as,

For inverse function of g,

g = {(- 7, 7), (- 6, - 7), (- 1, 6), (3, 9)

g⁻¹ = { (7, - 7), (- 7, - 6), (6, - 1), (9, 3) }

And, Inverse function of h is,

h (x) = (x + 13) / 5

y = (x + 13) / 5

Solve for x,

5y = x + 13

x = 5y - 13

h⁻¹ (x) = 5x - 13

So, We get;

⇒ (h⁻¹ о h ) (- 3)

⇒ (h⁻¹ (h (- 3))

⇒ (h⁻¹ (- 3 + 13)/5)

⇒ (h⁻¹ (2))

⇒ 5 (2) - 13

⇒ 10 - 13

⇒ - 3

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Answer:

x = 12

Step-by-step explanation:

sin45 = x/17         (sin = opposite/hypotenuse)

x = (sin45)(17) = 12.02 ≈ 12

The quadratic equation with roots 5 + i√2 and 5 - i√2 is:

x^2 - 10x + 27 = 0

To write a quadratic equation with roots 5 + i√2 and 5 - i√2, we can use the fact that complex roots occur in conjugate pairs. Therefore, the equation will have the form:

(x - root1)(x - root2) = 0

Substituting the given roots:

(x - (5 + i√2))(x - (5 - i√2)) = 0

Now, we expand the equation:

(x - 5 - i√2)(x - 5 + i√2) = 0

Using the difference of squares formula:

((x - 5)^2 - (i√2)^2) = 0

Simplifying the equation:

(x - 5)^2 + 2 = 0

Expanding the square:

x^2 - 10x + 25 + 2 = 0

Combining like terms:

x^2 - 10x + 27 = 0

Therefore, the quadratic equation with roots 5 + i√2 and 5 - i√2 is:

x^2 - 10x + 27 = 0

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Incorrect,  the correct exponential growth equation as it accurately represents a starting value of 300 and a growth rate of 2%. Scott's equation

Neither Pierre nor Scott has the correct exponential growth equation.

The exponential growth equation represents a relationship where a quantity increases or grows exponentially over time. It is typically represented as y = a(1 + r)^x, where "a" represents the initial or starting value, "r" represents the growth rate (expressed as a decimal), "x" represents the time or number of periods, and "y" represents the resulting value after the growth.

In this case, Pierre's equation is y =[tex]300(1.02)^x.[/tex]This equation suggests a growth rate of 2% (0.02 as a decimal), which means that the quantity would increase by 2% with each period. This aligns with the given growth rate of 2%. Thus, Pierre's equation is correct.

On the other hand, Scott's equation is y = [tex]300(1.2)^x[/tex]. This equation suggests a growth rate of 20% (0.2 as a decimal), which means that the quantity would increase by 20% with each period. However, the given growth rate is 2%, not 20%. Therefore, Scott's equation is incorrect.

To summarize, Pierre's equation, y = 300(1.02)^x, is the correct exponential growth equation as it accurately represents a starting value of 300 and a growth rate of 2%. Scott's equation, y = 300(1.2)^x, does not match the given growth rate and is therefore incorrect.

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By using the table of values, a graph of the exponential function is shown in the image below.

In Mathematics and Geometry, an exponential function can be modeled by using this mathematical equation:

[tex]f(x) = a(b)^x[/tex]

Where:

Based on the information provided above, we can logically deduce the following exponential function;

[tex]y = 2^x[/tex]

Next, we would create a table of values based on the exponential function;

when x = 0, the y-value is given by;

y = 2⁰

y = 1

when x = 1, the y-value is given by;

y = 2¹

y = 2

x                     y____

-2                 0.25

-1                  0.5

0                   1

1                    2

2                   4

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The solution set to the system of inequalities will be the overlapping region or the intersection of the shaded regions from all three inequalities.

The system of inequalities consists of three inequalities:

y ≤ 2x + 2

(1/2)x + y < 7

y - 3 ≥ 0

Let's analyze each inequality:

y ≤ 2x + 2 represents a shaded region below the line with a slope of 2 and a y-intercept of 2.

(1/2)x + y < 7 represents a shaded region below the line with a slope of -1/2 and a y-intercept of 7.

y - 3 ≥ 0 represents a shaded region above the line with a slope of 0 and a y-intercept of 3.

The solution set to the system of inequalities will be the overlapping region or the intersection of the shaded regions from all three inequalities.

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Answer:

Step-by-step explanation: On the left side find the number that is able to make that sum true for that equation. on the right side you just subtract the answer with the number to get your answer.

Answer:

Step-by-step explanation:

I Think this is the  answer

On the left side find the number that is able to make that sum true for that equation. on the right side you just subtract the answer with the number to get your answer

Miguel has 3 friends and he packed sandwiches for everyone to share equally means that Miguel packed 3 sandwiches for each camper.

If Miguel packed 4 sandwiches, then he would have packed 1 sandwich for each camper:

Number of sandwiches per camper = Total number of sandwiches / Number of campers

There are 4 campers, so plug that into the equation to get:

Number of sandwiches per camper = Total number of sandwiches / 4

Solve for the total number of sandwiches by multiplying both sides of the equation by 4:

Total number of sandwiches = Number of sandwiches per camper × 4

So, the total number of sandwiches is 4 × Number of sandwiches per camper.

Each camper will get 1 sandwich, so plug that into the equation to get:

Total number of sandwiches = 1 × 4

Which means there are a total of 4 sandwiches.

Therefore, Miguel packed 1 sandwich for each camper.

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Answer: The value of x = 1  and y = -1

Step-by Step Explanation:

We have 5x + 2y = 3 -----(i)

and 2x + 3y = -1 -----(ii)

By substitution method,

from (i), x = 3-2y/5

Putting the value of x in equation (ii),

we get, 2(3-2y/5) + 3y = -1

6 - 4y/5 + 3y =-1

6 - 4y + 15y = -5

6 - 11y = -5

-11y = -5 - 6

-11y = -11

y = -1

And, x = 3-2(-1)/5

x = 3+2/5

x = 1

Therefore,  x=1 and y=−1

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a. The total length of the painted white line in the javelin throwing arena is approximately 255.984 meters.

b. The total area of grass in the shaded landing area of the javelin throwing arena is approximately 1624.6 square meters.

To find the total length of the painted white line, we need to calculate the length of the two straight segments and the two circular arcs.

a) Total length of the painted white line:

Let's break it down into components:

1. The two straight segments: Each straight segment is 95 meters long, and there are two of them.

Length of straight segments = 2 * 95 = 190 meters

2. The two circular arcs:

The throwing arena is a sector of a circle with a radius of 100 meters. The angle of the sector can be calculated using trigonometry. The angle can be found by taking the inverse cosine of the ratio of the adjacent side (which is the radius minus the throwing line distance) to the hypotenuse (which is the radius).

Angle (in radians) = cos⁻¹((radius - throwing line distance) / radius)

Now, the length of each circular arc can be calculated using the formula for the length of an arc of a circle:

Length of circular arc = radius * angle

Let's calculate the angle first:

Angle (in radians) = cos⁻¹((100 - 5) / 100)

Angle = cos⁻¹(95 / 100)

Angle ≈ 0.32492 radians

Now, we can calculate the length of each circular arc:

Length of each circular arc = 100 * 0.32492 ≈ 32.492 meters

Since there are two circular arcs, the total length of the painted white line is:

Total length = 190 (straight segments) + 2 * 32.492 (circular arcs)

Total length ≈ 255.984 meters

b) Total area of grass in the shaded landing area:

The shaded landing area is the sector of the circle with a radius of 100 meters and the angle we calculated above.

The formula to calculate the area of a sector of a circle is:

Area of sector = (angle / 2π) * π * radius²

Area of the shaded landing area = (0.32492 / (2π)) * π * 100²

Area of the shaded landing area ≈ (0.32492 / 2) * 10000

Area of the shaded landing area ≈ 1624.6 square meters

So, the total area of grass in the shaded landing area is approximately 1624.6 square meters.

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The average utility price in Orlando for the 6 month period, and the way it compares to the national average is d. $ 308. 83 ; higher than the national average .

The information for Orlando is April, the cost was 288 dollars; May, 310 dollars; June, 325 dollars; July, 294 dollars; August, 293 dollars; September, 343 dollars.

The average is:

= ( 288 + 310 + 325 + 294 + 293 + 343 ) / 6

= 1, 853 / 6

= $ 308. 83

This average is higher than the national average of $ 308. 83.

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Full question is:

The average national utility price is $270.48. Over a 6-month period, what is the average utility price in Orlando? How does this compare with the national average?

A graph titled Orlando, Florida, has month on the x-axis and utility price on the y-axis. In April, the cost was 288 dollars; May, 310 dollars; June, 325 dollars; July, 294 dollars; August, 293 dollars; September, 343 dollars.

a. $370.60; higher than the national average

b. $292.17; higher than the national average

c. $38.35; lower than the national average

d. $308.83; higher than the national average

Answer:

1/4+1/3+1/8

=17/24

Step-by-step explanation:

1-17/24

=7/24

7/24×240

=70

Answer:

  y < 2x +6

Step-by-step explanation:

You want an inequality for the given graph.

Here are some steps you can follow, in no particular order.

When you have this information, you can write the inequality in slope-intercept form.

When the boundary line is dashed, the inequality symbol you use will not include the "or equal to" case. It will be one of < or >.

When the shading is below the line, the values of y that satisfy the inequality will be less than (<) those on the boundary line. If shading is above, the y-values will be greater than (>) those on the line.

The slope and intercept go into the inequality like this:

  y < mx + b . . . . . . where m is the slope, and b is the y-intercept

For a dashed line, shaded below, with m=2 and b=6, the inequality is ...

  y < 2x +6

__

Additional comment

There are two points identified on the boundary line: (-2, 2) and (0, 6). The slope formula can be used to find the slope:

  m = (y2 -y1)/(x2 -x1)

  m = (6 -2)/(0 -(-2)) = 4/2 = 2

The point (0, 6) on the y-axis is the y-intercept. The y-value there is 6.

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Answer: There are 16 fluid ounces in 1 pint. To determine the number of pints in the jug, we need to divide the total number of fluid ounces by 16.

Given that the jug contains 36 fluid ounces of apple juice, we divide 36 by 16:

36 fluid ounces ÷ 16 fluid ounces/pint = 2.25 pints

Therefore, the jug contains 2.25 pints of apple juice.

The maximum number of CYN Dewi can buy is 4268.8 CYN and cost in pounds is  £453.20.

Maximum number of CYN Dewi can buy:

Since £1 buys 9.28 CYN, Dewi's £460 can be converted to Chinese yuan by multiplying it by the exchange rate:

Maximum CYN = £460 × 9.28 CYN/£1

To calculate this, we multiply the pound amount by the exchange rate:

Maximum CYN = £460 × 9.28 CYN/£1 ≈ 4268.8 CYN

Cost in pounds to the nearest penny:

To calculate the cost in pounds, we divide the desired amount of Chinese yuan by the exchange rate:

Cost in pounds = 4268.8 CYN ÷ 9.42 CYN/£1

To calculate this, we divide the Chinese yuan amount by the exchange rate:

Cost in pounds = 4268.8 CYN ÷ 9.42 CYN/£1

= £453.20

Therefore, the cost in pounds, to the nearest penny, will be £453.20.

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Complete question

Buying chinese yuan (CYN)             £1 buys 9.28 CYN

Selling chinese yuan (CYN)           9.42 CYN buys  £1

(a) Dewi has £460 to buy Chinese yuan.

Calculate the maximum number of CYN Dewi can buy, and

how much, to the nearest penny, this will cost him.

The equation of the line with a slope of 1/4 that passes through the point (3,0) is y = 1/4x - 3/4.

To find the equation of a line with a slope of 1/4 that passes through the point (3,0), we can use the point-slope form of the equation of a line.

The point-slope form of a line is:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope.

Substituting the values into the formula, we have:

y - 0 = 1/4(x - 3)

Simplifying:

y = 1/4(x - 3)

Distributing 1/4 throughout the expression:

y = 1/4x - 3/4

Therefore, the equation of the line with a slope of 1/4 that passes through the point (3,0) is y = 1/4x - 3/4.

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A square and a traingle are present in a large rectangle with given dimensions in the figure.

[tex]\qquad\displaystyle \tt \dashrightarrow \: 12 \times 8[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 96 \: \: unit {}^{2} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 4 \times 4[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 16 \: \: unit {}^{2} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 2 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 10 \: \: unit {}^{2} [/tex]

[ area of square / total area ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{16}{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{1}{6} [/tex]

[ total area except area of triangle / total area ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{96 - 10}{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{86 }{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{43}{48} [/tex]

Answer:  if the length of the second rectangular field is 200 meters, the width should be 35 meters to have the same perimeter but a larger area.

Step-by-step explanation:

STEP1:- Let's denote the length of the second rectangular field as L2 and the width as W2.

The perimeter of a rectangle is given by the formula:

Perimeter = 2(length + width).

For the first rectangular field with length L1 = 135 meters and width W1 = 100 meters, the perimeter is:

Perimeter1 = 2(135 + 100) = 470 meters.

STEP 2:- To find the length and width of the second rectangular field with the same perimeter but a larger area, we need to consider that the perimeters of both rectangles are equal.

Perimeter1 = Perimeter2

470 = 2(L2 + W2)

STEP 3 :- To determine the larger area, we need to find the corresponding length and width. However, there are multiple solutions for this problem. We can set an arbitrary value for one of the dimensions and calculate the other.

For example, let's assume the length of the second rectangular field as L2 = 200 meters:

470 = 2(200 + W2)

470 = 400 + 2W2

2W2 = 470 - 400

2W2 = 70

W2 = 35 meters

HENCE L2 = 200 meters and W2 = 35 meters