Geometry
three squares with areas of 64, 225, and 289 square units are arranged so that when their vertices coincide a triangle is formed. find the area of that triangle.

please explain how you solved this along with the answer.

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{7})\hspace{10em} \stackrel{slope}{m} ~=~ - 7 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{- 7}(x-\stackrel{x_1}{2}) \\\\\\ y-7=-7x+14\implies {\Large \begin{array}{llll} y=-7x+21 \end{array}}[/tex]

Answer:

C) [tex]\sf \sqrt{45}[/tex]

Step-by-step explanation:

Pythagorean theorem:

               AB = 6 units

              BC = 3 units

AC is hypotenuse and AB is the base and BC is the altitude.

Hypotenuse² = base² + altitude²

               AC² = AB² + BC²

                      [tex]\sf = 6^2 + 3^2\\\\ = 36 + 9\\\\ = 45[/tex]

               [tex]\sf AC= \sqrt{45}[/tex]

The absolute maximum of f(x) = 4x / (x² + 1) on the interval [-4,0] occurs at x = 2 and the absolute minimum occurs at x = -4.

To find the absolute extrema, we first find the critical points by setting the derivative of f(x) equal to zero:

f'(x) = (4(x² + 1) - 8x²) / (x² + 1)² = 0

Simplifying, we get:

4 - 4x² = 0

x² = 1

x = ±1

Since x = -4 and x = 0 are also endpoints of the interval, we evaluate f(x) at these five points:

f(-4) = -8/17

f(-1) = -4/5

f(0) = 0

f(1) = 4/5

f(2) = 8/5

Thus, the absolute maximum occurs at x = 2, where f(x) = 8/5, and the absolute minimum occurs at x = -4, where f(x) = -8/17.

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The work done in moving the particle from the origin to x = 4 under the influence of the force F (x) = 8[tex]x^3[/tex]-5 is 492 units of work.

The work done in moving a particle along a path under the influence of a force, we use the work-energy principle.

This principle states that the work done on a particle by a force is equal to the change in the particle's kinetic energy.

Mathematically this can be expressed as:

W = ΔK

Where

W is the work done,

ΔK is the change in kinetic energy and

Both are scalar quantities.

The work done by a force on a particle along a path is given by the line integral:

W = ∫ C F · ds

Where,

C is the path,

F is the force,

ds is the differential displacement along the path and  denotes the dot product.

In the case where the force is a function of position only (i.e., F = F(x,y,z)), we can evaluate the line integral using the parametric equations for the path.

If the path is given by the parameterization r(t) = <x(t), y(t), z(t)>, then we have:

W = ∫ [tex]a^b[/tex] F(r(t)) · r'(t) dt

The work done in moving the particle from the origin to a final position at x = 4. We can evaluate the work done using the definite integral of the force from x = 0 to x = 4, as shown in the solution.

The initial kinetic energy is zero.

The work done by the force in moving the particle from x = 0 to x = 4 is given by the definite integral:

W = ∫ F(x) dx

Substituting the given expression for the force, we have:

W = ∫0 (8x - 5) dx

Integrating with respect to x, we have:

W = [(2x - 5x)]_0

W = (2(4) - 5(4)) - (2(0) - 5(0))

W = 512 - 20

W = 492

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Answer:

Set your calculator to degree mode.

Using the Law of Sines:

7/sin(40°) = x/sin(81°)

x = 7sin(81°)/sin(40°) = 10.8

Answer:

10.8=x

Step-by-step explanation:

Using the Law of Sines, we can put together the fact that

[tex]\frac{sin A}{a} =\frac{sinB}{b}[/tex]

Substitute our given values from the triangle:

[tex]\frac{sin 81}{x} =\frac{sin40}{7}[/tex]

Turn the sines into a decimal:

[tex]\frac{0.9876}{x} =\frac{0.6427}{7}\\[/tex]

cross multiply using butterfly method

0.988·7=0.643x

solve for x

6.916=0.643x

divide both sides by 0.643

10.8=x (round to nearest tenth)

Hope this helps! :)

The missing values of the table representing the number of miles driven is proportional to the number of gallons consumed are,

Gallons consumed :    2         4             7          8           10          12

Miles driven            :    54        108        189      216        270       324

Let us consider 'x' represents the number of miles driven .

And 'y' represents the number of gallons consumed .

Number of miles driven is proportional to the number of gallons consumed.

⇒ ( x₁ / x₂ ) = ( y₁ / y₂ )

⇒ ( 2/ 4) = ( 54 / y₂ )

⇒y₂  = ( 54 × 4 ) / 2

⇒y₂  = 108.

⇒ ( x₁ / x₃ ) = ( y₁ / y₃ )

⇒ ( 2/ x₃) = ( 54 / 189)

⇒x₃  = ( 189 × 2 ) / 54

⇒x₃= 7

⇒ ( x₁ / x₅ ) = ( y₁ / y₅ )

⇒ ( 2/ 10) = ( 54 / y₅ )

⇒y₅  = ( 54 × 10 ) / 2

⇒y₅  = 270.

⇒ ( x₁ / x₆ ) = ( y₁ / y₆ )

⇒ ( 2/ 12) = ( 54 / y₆ )

⇒y₆ = ( 54 × 12 ) / 2

⇒y₆  = 324

Therefore, using the proportional relation between miles driven and gallons consumed the value of the table are,

Gallons consumed :    2         4             7          8           10          12

Miles driven            :    54        108        189      216        270       324

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The above question is incomplete, the complete question is:

Randy is planning to drive from New Jersey to Florida. Every time Randy stops for gas, he records the distance he traveled in miles and the total number of gallons used. Assume that the number of miles driven is proportional to the number of gallons consumed in order to complete the table.

Gallons consumed :    2         4                    8           10       12

Miles driven            :    54                189       216        

The profitability index is 0.1237.

To find the profitability index (PI), we need to divide the present value of the cash flows by the initial investment.

To calculate the present value of the cash flows, we need to discount each cash flow to its present value and then add them up. Using a discount rate of 10%, we get:

Year 0: -$28,500 / [tex](1 + 0.10)^0[/tex]= -$28,500

Year 1: $15,200 /[tex](1 + 0.10)^1[/tex]= $13,818.18

Year 2: $13,700 / [tex](1 + 0.10)^2[/tex] = $10,881.68

Year 3: $10,100 /[tex](1 + 0.10)^3[/tex] = $7,322.51

The sum of the present values is:

PV = -$28,500 + $13,818.18 + $10,881.68 + $7,322.51 =

PV = $3,521.37

The profitability index is therefore:

PI = PV / Initial Investment = $3,521.37 / $28,500 = 0.1237

So the profitability index is 0.1237.

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When we multiply (2x - 1) and (x + 2) using FOIL method, we get:

(2x - 1)(x + 2) = 2x(x) + 2x(2) - 1(x) - 1(2)

= 2x² + 4x - x - 2

= 2x² + 3x - 2

Therefore, the product of (2x - 1) and (x + 2) is 2x² + 3x - 2.

An object moves in simple harmonic motion with a period of 8 minutes and an amplitude of 12 m. At time =t0 minutes, its displacement d from rest is −12m, and initially, it moves in a positive direction. We can write the final equation for the displacement d as a function of time t: d(t) = 12 * cos((π/4)t + π)

To model the displacement d as a function of time t for an object in simple harmonic motion with a period of 8 minutes and an amplitude of 12m, we'll use the following equation:

d(t) = A * cos(ωt + φ)

where:
- d(t) is the displacement at time t
- A is the amplitude (12m in this case)
- ω is the angular frequency, calculated as (2π / period)
- t is the time in minutes
- φ is the phase angle, which we'll determine based on the initial conditions

Since the period is 8 minutes, we can calculate the angular frequency as follows:
ω = (2π / 8) = (π / 4)

At t = 0 minutes, the displacement is -12m, and the object moves in a positive direction. So we have:
-12 = 12 * cos(φ)

Dividing both sides by 12:
-1 = cos(φ)

Therefore, φ = π (or 180°) since the cosine of π is -1.

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Answer:

The given equation Y=4x^3+6 is a nonlinear equation because it contains a term with a power of 3, which means that the relationship between Y and x is not linear. In a linear equation, the power of the variable is always 1. Therefore, the answer is B. Nonlinear.

Step-by-step explanation:

The prediction that is not supported by the data is option B: "The number of non-fiction books sold on Friday will be two-and-a-half times the number of arts and crafts books sold on Friday."

We can see from the table that on Thursday, 7 sports and trivia books and 19 arts and crafts books were sold, for a difference of 12.

On Thursday, 13 non-fiction books and 19 arts and crafts books were sold. If we assume that the same ratio will hold on Friday, then we can predict that the number of non-fiction books sold will be (19/2)*2.5 = 23.75, which is not a whole number. Therefore, this prediction is not supported by the data.

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The function f(x) = (¹/₂)x⁴ +2x³ is concave up when f''(x) > 0, which is true when x > 0 or x < -2.

The concavity of a function is determined by taking the second derivative.

f'(x) = 2x³ + 6x²

f''(x) = 6x² + 12x

To find out when f(x) is concave up, we need to determine when f''(x) is positive;

f''(x) > 0

6x² + 12x > 0

6x(x + 2) > 0

When x > 0, both factors are positive, and the inequality is true.

When x < -2, both factors are negative, and the inequality is true.

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If the circumference of a circle is 50. 4 ft, its area is 202.24 sq ft. Correct  option is C: 202.24 sq ft.

The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. We are given that the circumference is 50.4 ft, so we can solve for the radius:

50.4 = 2πr

r = 50.4 / (2π)

r ≈ 8.02 ft

Now we can use the formula for the area of a circle: A = πr²

A = 3.14 * (8.02)²

A ≈ 202.24 sq ft

Therefore, the answer is option C: 202.24 sq ft.

Alternatively, to find the area of a circle with a circumference of 50.4 ft, we will first find the radius using the formula for circumference (C = 2πr) and then use the formula for the area of a circle (A = πr²). Using π = 3.14:

Solve for the radius (r):
C = 2πr
50.4 = 2(3.14)r
r = 50.4 / (2 * 3.14)
r ≈ 8 ft

Calculate the area (A):
A = πr²
A = 3.14 * (8²)
A = 3.14 * 64
A ≈ 201.06 sq ft

The closest answer among the options provided is 202.24 sq ft. Correct  option is C: 202.24 sq ft.

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Entonces, 1/8 de su salario se va en renta, y su salario mensual es de $1200.

Entiendo que deseas saber qué fracción del salario de Edna se va en renta y cuál es su salario mensual si su renta mensual es de $150. Para resolver esto, seguiré estos pasos:

1. Sabemos que Edna gasta la mitad de su salario en renta, utilidades y comida. Esto se puede expresar como 1/2 de su salario.
2. Se nos dice que 1/4 de esos gastos se va en renta. Entonces, para encontrar la fracción de su salario que se va en renta, multiplicamos 1/2 (gastos totales) por 1/4 (fracción de gastos en renta): (1/2) * (1/4) = 1/8.
3. Ahora sabemos que 1/8 de su salario se va en renta. Para hallar su salario mensual, utilizaremos la renta mensual de $150.
4. Si 1/8 de su salario equivale a $150, podemos resolver su salario (S) utilizando la siguiente ecuación: (1/8)S = 150.
5. Para encontrar S, multiplicamos ambos lados de la ecuación por 8: S = 150 * 8.
6. Finalmente, calculamos el valor: S = 1200.

Entonces, 1/8 de su salario se va en renta, y su salario mensual es de $1200.

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The 10 breeding pairs of spotted owls in Cascade National Park studied by the researcher represent a selected sample. So, the correct answer is A)

The 10 breeding pairs of spotted owls represent a selected sample.

A sample is a subset of a larger population that is chosen for research or study purposes. In this case, the researcher is interested in studying the disappearance of spotted owls from northwestern forests, but it would be impractical to study the entire population of spotted owls in the region.

Therefore, the researcher selected a smaller group of 10 breeding pairs in Cascade National Park as a representative sample to study over the course of one year. The selected sample may help the researcher to draw conclusions about the population of spotted owls in the region. So, the correct option is A).

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--The given question is incomplete, the complete question is given

" a researcher is interested in the disappearance of spotted owls from northwestern forests. she studies 10 breeding pairs of spotted owls in cascade national park for one year. the 10 breeding pairs are the: group of answer choices

selected sample

snowball sample  

stratified sample

target population"--

Answer:

1/4

Step-by-step explanation:

The probability of flipping tails is 1/2, since there are two equally likely outcomes when flipping a coin (heads or tails).

The probability of rolling an odd number on a standard number cube is 3/6 or 1/2, since there are three odd numbers (1, 3, and 5) out of six possible outcomes (1, 2, 3, 4, 5, and 6).

To find the probability of both events happening (i.e., flipping tails and rolling an odd number), we multiply the probabilities of each event:

P(tails and odd number) = P(tails) * P(odd number)

P(tails and odd number) = 1/2 * 1/2

P(tails and odd number) = 1/4

Therefore, the probability of flipping tails and rolling an odd number is 1/4 or 0.25.

The closest option to this value is d) 10.75, but none of the options is an exact match.

To find the interquartile range (IQR) of the data, we need to first find the first quartile (Q1) and the third quartile (Q3).

To do this, we can arrange the data in order from smallest to largest:

98, 100, 104, 105, 106, 110, 112, 115, 116, 118, 120, 122, 126, 130, 136, 140

The median of the data is the average of the two middle values, which are 112 and 115. So, the median is (112 + 115) / 2 = 113.5.

To find Q1, we need to find the median of the data values below the median. These are:

98, 100, 104, 105, 106, 110, 112, 115

The median of these values is (106 + 110) / 2 = 108.

To find Q3, we need to find the median of the data values above the median. These are:

116, 118, 120, 122, 126, 130, 136, 140

The median of these values is (122 + 126) / 2 = 124.

Now we can calculate the interquartile range (IQR) as the difference between Q3 and Q1:

IQR = Q3 - Q1 = 124 - 108 = 16.

Therefore, the interquartile range of the data is 16, or in decimals 16.00.

The closest option to this value is d) 10.75, but none of the options is an exact match.

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Afia visits the shopping mall on Tuesday to purchase some groceries. if she goes back after 295 days, she visits the shopping mall again​ on a Wednesday.

To find out what day Afia visited the shopping mall again, we can divide  295 by 7  because there are 7 days in a week. we need to find out how many full weeks have passed and how many days.

= 295/ 7 = 42.1

The 295 divided by 7 is 42 with a remainder of 1 or we can write as that 42 full weeks and 1 day have passed.

When 42 weeks have passed that day will be Tuesday and the 1 day after Tuesday is Wednesday.

Therefore,  Afia visited the shopping mall again on a Wednesday.

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Answer:

Ans=28

Step-by-step explanation:

ƒ(x) = 6x2 - 3x + 22xf( - 2)=[?]

at ƒ(-2)

Substitute each x with -2

ƒ(-2) = 6(-2)2 - 3(-2) - 2

ƒ(-2) = 6(4) - 3(-2) - 2

ƒ(-2) = 24 + 6 + 0 - 2

ƒ(-2) = 28

I hope I was right

The probability of Akhil picking two oranges is 11/105 OR 10.476%.

To find the probability of Akhil picking two oranges, we need to consider the total number of fruits and the number of oranges in the basket.

The total number of fruits is 8 green apples + 11 red apples + 5 nectarines + 12 oranges = 36 fruits.

The probability of picking the first orange is 12 oranges / 36 fruits = 1/3.

After eating the first orange, there are now 35 fruits left and 11 oranges.

The probability of picking the second orange is 11 oranges / 35 fruits.

So, the probability of Akhil picking two oranges (P(two oranges)) is the product of the individual probabilities: (1/3) * (11/35) = 11/105.

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|r| = 7/80 < 1, the series is convergent. The sum = 0/(1-7/80) = 0. the sum of the geometric series is 0.

The geometric series with first term 0 and common ratio 7/80 is given by 0, 7/80, (7/80)², (7/80)³, ... In general, the nth term is (7/80)ⁿ⁻¹.

To determine whether this series is convergent or divergent, we can use the formula for the sum of an infinite geometric series:

sum = a/(1-r)

where a is the first term and r is the common ratio. In this case, a = 0 and r = 7/80.

If |r| < 1, then the series converges to the sum given by the above formula. If |r| ≥ 1, then the series diverges.

In this case, |r| = 7/80 < 1, so the series is convergent. The sum is given by:

sum = 0/(1-7/80) = 0

Therefore, the sum of the geometric series is 0.

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The area of the commune's summer-autumn rice crop is approximately 570,041.902 square meters. ( rounded to the third decimal place)

To calculate the area of the commune's summer-autumn rice crop, we need to multiply the total area of the commune by the percentage of the area used to grow rice.

First, we need to convert the area from hectares to square meters since the percentage is based on the total land area in square meters.

1 hectare = 10,000 square meters

So, 81.5 hectares = 81.5 x 10,000 = 815,000 square meters

Next, we can calculate the area of the rice crop:

57/81.5 = 0.699386503

0.699386503 x 815,000 = 570,041.902 square meters

Rounding to the third decimal place, the area of the commune's summer-autumn rice crop is approximately 570,041.902 square meters.

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The measure of each arc in the circle is given by: 360 degrees / n

where n= number of spokes

If the spokes on a bicycle wheel divide the wheel into congruent sections, then each section is an equal angle at the center of the circle. Since there are "n" spokes on the wheel, the circle will be divided into "n" congruent sections.

Therefore, the measure of each arc in the circle is given by:

= 360 degrees / n

For example, if there are 18 spokes on the wheel, then each arc will have a measure of:

360 degrees / 18 = 20 degrees

So each arc would measure 20 degrees.

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The objects used to measure mass are Balances and Scales. The objects used to measure Volume are Manometer, Pipettes, Graduated cylinders, Flasks, Volumetric flasks and Beakers. Here Density formula can be used to measure both mass and volume.

The problem is asking to match different measuring tools with the measurements they are used for, i.e., mass or volume.

The first tool is a manometer. A manometer is used to measure pressure and not mass or volume, so it does not belong in either category.

The next set of tools are pipettes, graduated cylinders, and volumetric flasks. These tools are all used to measure volume, so they belong in the volume category.

The next set of tools are scales and balances. These tools are used to measure mass, so they belong in the mass category.

The formula for density can be used to calculate the mass of an object given its volume and density, or the volume of an object given its mass and density, so it belongs in both categories.

Finally, a beaker or a graduated cylinder can be used to measure volume, so it belongs in the volume category.

Therefore, the correct categorization of the measuring tools are as follows

Volume

Pipettes

Graduated cylinders

Volumetric flasks

Beaker or graduated cylinder

Mass

Scales

balances

Both

Formula for density

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The arc length of the polar curve r = 8θ^2 on the interval [0, 2] is approximately 70.71 units.

The polar curve r = 8θ^2 on the interval [0, 2] has an arc length which can be calculated using the formula for arc length in polar coordinates:

L = ∫√(r^2 + (dr/dθ)^2) dθ, from θ = 0 to θ = 2.
First, we need to find the derivative dr/dθ:

r = 8θ^2, so dr/dθ = 16θ.

Now, plug r and dr/dθ into the arc length formula:
L = ∫√((8θ^2)^2 + (16θ)^2) dθ, from θ = 0 to θ = 2.

Simplify the integrand:

L = ∫√(64θ^4 + 256θ^2) dθ, from θ = 0 to θ = 2.

Factor out 64θ^2:

L = ∫√(64θ^2(1 + θ^2)) dθ, from θ = 0 to θ = 2.

Now, apply the substitution u = 1 + θ^2, so du = 2θ dθ:

L = 32∫√(u) du, from u = 1 to u = 5.

Integrate and evaluate:

L = (32/3)(u^(3/2)) | from u = 1 to u = 5.

L = (32/3)(5^(3/2) - 1^(3/2)).

L ≈ 70.71 units.

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The probability of picking a 20 dollar bill is 1/7 or 14.3%

The probability of a thing is the likelihood or number of chances that such a thing will occur. For the scenario given,

There are a total of 7 bills in your pocket

1, 1, 1, 1,

5, 5,

20.

To find the probability of picking out the twenty dollar bill, divide the number of twenty dollar bills by the the total of the number of bills you are with.

Probability = 1/ 7 which can be converted to % = 14.3%.

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The function represented by the data is f(1/4)x³ + 2x² - 24. The correct option is 2.

In the given table, we have the values of x and f(x) for x=0,1,2,3, and 4. We need to find a polynomial function that satisfies these data points.

Looking at the table, we can see that f(x) is negative for x=0,1,2 and positive for x=3,4. This suggests that the polynomial has a root or a zero between x=2 and x=3.

To find the degree of the polynomial, we count the number of data points given. Since we have 5 data points, we need a polynomial of degree 4.

We can use interpolation to find the coefficients of the polynomial. One way to do this is to set up a system of equations using the data points:

f(0) = -24 = a(0)⁴ + b(0)³ + c(0)² + d(0) + e

f(1) = -21.75 = a(1)⁴ + b(1)³ + c(1)² + d(1) + e

f(2) = -14 = a(2)⁴ + b(2)³ + c(2)² + d(2) + e

f(3) = 0.75 = a(3)⁴ + b(3)³ + c(3)² + d(3) + e

f(4) = 24 = a(4)⁴ + b(4)³ + c(4)² + d(4) + e

Solving this system of equations gives us the polynomial function:

f(x) = -0.25x⁴ + 2x³ - 2.75x² - 0.5x + 24

Therefore, the correct option is 2. f(x) = (1/4)x³ + 2x² - 24.

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At point (3, 9), the rate of change of temperature with respect to the x-axis is -4.2 °C/m, and with respect to the y-axis, it is -23.4 °C/m.

To find the rates of change of the temperature with respect to the distances moved along the x- and y-axes at point (3, 9), you need to compute the partial derivatives of the temperature function T(x, y) = 900 - 0.7x^2 - 1.3y^2 with respect to x and y.
For the x-axis:
∂T/∂x = -1.4x
At point (3, 9), ∂T/∂x = -1.4(3) = -4.2 °C/m
For the y-axis:
∂T/∂y = -2.6y
At point (3, 9), ∂T/∂y = -2.6(9) = -23.4 °C/m
So, at point (3, 9), the rate of change of temperature with respect to the x-axis is -4.2 °C/m, and with respect to the y-axis, it is -23.4 °C/m.

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To calculate the percentage of one's monthly budget that transportation costs account for, we need to know the total amount of money spent on transportation and the total monthly budget.

Let's say, for example, that John spends $500 per month on transportation and his monthly budget is $2,000.

To calculate the percentage, we would divide the amount spent on transportation by the total monthly budget and then multiply by 100 to get the percentage. So, in this case, the calculation would be:

[tex]($500 / $2,000) x 100 = 25%[/tex]

Therefore, John's transportation costs account for 25% of his monthly budget. This is a significant portion of his budget, and if he needs to save money, he may want to consider alternative modes of transportation such as carpooling,

public transportation, or biking. It's always important to keep track of expenses and prioritize spending in order to maintain a healthy financial situation.

To know more about transportation costs refer here

https://brainly.com/question/29544207#

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