1. a family bought a new car for a purchase price of $32,000. the car will lose 15% of its value the day it is purchased and the car will depreciate at a constant rate following that. the value of the car as a function of time can be modeled by ( y = c - 0.09cx ), where ( y ) is the value of the car ( x ) years after the car was purchased and ( c ) is the value of the car after the initial 15% depreciation.

a. what is the value of the car 2 years after its purchase date? show your work.
b. on an ( xy )-grid, graph the value, ( y ), of the car, as a function of ( x ), where ( x ) represents the number of years after the purchase date, for ( 0 leq x leq 7 ) years. label the axes and show the scales used for the graph.
c. use your graph to estimate the number of years, ( x ), after the purchase date that the value of the car is $15,000. label this point on your graph and indicate the approximate coordinates of the point.
d. algebraically find the number of years, ( x ), after the purchase date that the value of the car is exactly $15,000. round your solution to the nearest tenth of a year. show your work.

2.

the diagram above shows the plan berenice has for a triangular splash pad in a local city park. there will be three circles, each with a diameter of 6 feet, and the circles will be enclosed by an equilateral triangle with a side length of 30 feet. berenice plans to have splash areas/fountains within the circles and walkways in the shaded areas of the triangle.
a. according to the diagram, what is the maximum area, in square feet, available for berenice to have for splash/fountain areas in the triangle? show your work.
b. according to the diagram, what percentage of the triangle will be set aside for walkways? show your work.
c. if a ball were to fall on a random point within the triangular splash pad, what is the probability that the ball would fall where berenice plans to have splash areas? show your work or explain your reasoning.

Question

1. a family bought a new car for a purchase price of $32,000. the car will lose 15% of its value the day it is purchased and the car will depreciate at a constant rate following that. the value of the car as a function of time can be modeled by ( y = c - 0.09cx ), where ( y ) is the value of the car ( x ) years after the car was purchased and ( c ) is the value of the car after the initial 15% depreciation.

a. what is the value of the car 2 years after its purchase date? show your work.
   b. on an ( xy )-grid, graph the value, ( y ), of the car, as a function of ( x ), where ( x ) represents the number of years after the purchase date, for ( 0 leq x leq 7 ) years. label the axes and show the scales used for the graph.
   c. use your graph to estimate the number of years, ( x ), after the purchase date that the value of the car is $15,000. label this point on your graph and indicate the approximate coordinates of the point.
   d. algebraically find the number of years, ( x ), after the purchase date that the value of the car is exactly $15,000. round your solution to the nearest tenth of a year. show your work.

2.

the diagram above shows the plan berenice has for a triangular splash pad in a local city park. there will be three circles, each with a diameter of 6 feet, and the circles will be enclosed by an equilateral triangle with a side length of 30 feet. berenice plans to have splash areas/fountains within the circles and walkways in the shaded areas of the triangle.
   a. according to the diagram, what is the maximum area, in square feet, available for berenice to have for splash/fountain areas in the triangle? show your work.
   b. according to the diagram, what percentage of the triangle will be set aside for walkways? show your work.
   c. if a ball were to fall on a random point within the triangular splash pad, what is the probability that the ball would fall where berenice plans to have splash areas? show your work or explain your reasoning.

Accepted Answer

### Problem 1 - Part A
# Explanation:
## Step1: Find the initial value $ c $
The car's purchase price is $32,000, and it loses 15% initially. So, $ c = 32000\times(1 - 0.15) $
$ c = 32000\times0.85 = 27200 $

## Step2: Use the function $ y = c - 0.09cx $ for $ x = 2 $
Substitute $ c = 27200 $ and $ x = 2 $ into the function:
$ y = 27200 - 0.09\times27200\times2 $
First, calculate $ 0.09\times27200 = 2448 $
Then, $ 2448\times2 = 4896 $
Finally, $ y = 27200 - 4896 = 22304 $

# Answer:
The value of the car 2 years after purchase is $22,304.

### Problem 1 - Part D
# Explanation:
## Step1: Set up the equation
We know $ y = 15000 $, $ c = 27200 $, and the function $ y = c - 0.09cx $. So:
$ 15000 = 27200 - 0.09\times27200\times x $

## Step2: Simplify the equation
First, calculate $ 0.09\times27200 = 2448 $, so the equation becomes:
$ 15000 = 27200 - 2448x $

## Step3: Solve for $ x $
Subtract 27200 from both sides:
$ 15000 - 27200 = -2448x $
$ -12200 = -2448x $
Divide both sides by -2448:
$ x = \frac{12200}{2448} \approx 4.98 \approx 5.0 $ (rounded to the nearest tenth)

# Answer:
The number of years is approximately 5.0 years.

### Problem 2 - Part A
# Explanation:
## Step1: Find the area of one circle
The diameter of each circle is 6 ft, so the radius $ r = \frac{6}{2} = 3 $ ft.
The area of a circle is $ A_{circle} = \pi r^2 $, so:
$ A_{circle} = \pi\times3^2 = 9\pi $ square feet.

## Step2: Find the total area of three circles
There are three circles, so total splash area $ A_{total} = 3\times9\pi = 27\pi \approx 27\times3.1416 \approx 84.82 $ square feet.

# Answer:
The maximum area for splash/fountain areas is approximately $ 27\pi $ (or about 84.82) square feet.

### Problem 2 - Part B
# Explanation:
## Step1: Find the area of the equilateral triangle
The side length of the equilateral triangle is 30 ft. The area of an equilateral triangle is $ A_{triangle} = \frac{\sqrt{3}}{4}s^2 $, where $ s = 30 $.
$ A_{triangle} = \frac{\sqrt{3}}{4}\times30^2 = \frac{\sqrt{3}}{4}\times900 = 225\sqrt{3} \approx 225\times1.732 \approx 389.7 $ square feet.

## Step2: Find the area of the walkways
Walkway area = Triangle area - Splash area. From Part A, splash area is $ 27\pi \approx 84.82 $
Walkway area $ \approx 389.7 - 84.82 = 304.88 $ square feet.

## Step3: Calculate the percentage
Percentage = $ \frac{\text{Walkway Area}}{\text{Triangle Area}}\times100 $
$ \text{Percentage} = \frac{304.88}{389.7}\times100 \approx 78.2\% $ (or using exact values: $ \frac{225\sqrt{3}-27\pi}{225\sqrt{3}}\times100 $)

# Answer:
Approximately 78.2% of the triangle is set aside for walkways (exact value can be calculated with more precision, but this is the approximate percentage).

### Problem 2 - Part C
# Explanation:
## Step1: Recall the areas
Splash area (from Part A) is $ 27\pi \approx 84.82 $ square feet.
Triangle area (from Part B) is $ 225\sqrt{3} \approx 389.7 $ square feet.

## Step2: Calculate the probability
Probability is the ratio of splash area to triangle area:
$ P = \frac{\text{Splash Area}}{\text{Triangle Area}} = \frac{27\pi}{225\sqrt{3}} $
Simplify: $ \frac{3\pi}{25\sqrt{3}} = \frac{\pi\sqrt{3}}{25} \approx \frac{3.1416\times1.732}{25} \approx \frac{5.441}{25} \approx 0.218 $ (or 21.8%)

# Answer:
The probability is approximately $ \frac{\pi\sqrt{3}}{25} $ (or about 0.218 or 21.8%).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1 - Part A

Step1: Find the initial value \( c \)

Step2: Use the function \( y = c - 0.09cx \) for \( x = 2 \)

Step1: Set up the equation

Step2: Simplify the equation

Step3: Solve for \( x \)

Step1: Find the area of one circle

Step2: Find the total area of three circles

Answer:

Problem 1 - Part D