1. dana is riding in a 100-mile-long bicycle race from indy to cincinnati. the function ( r(x) = 100 - 18x ) models the number of miles that dana has remaining ( x ) hours after she starts the race. what is the best interpretation of ( r(2) = 64 )?
a. dana rides about 64 miles in the first 2 hours of the race.
b. dana will ride the last 64 miles of the race in about 2 hours.
c. when dana has ridden 64 miles, she has about 2 hours remaining.
d. two hours after dana starts the race, she has about 64 miles remaining.

2. a wading pool that already contains 5 gallons of water is being filled. the function ( w(x) = 2.1x + 5 ) models the amount of water in the pool after ( x ) minutes. what does ( w(5) = 15.5 ) mean in the context of the problem?

3. the graph of the equation ( y = \frac{2}{3}x + c ), where ( c ) is a constant intersects the x-axis at ( (12, 0) ). what is the value of ( c )?

4. the function ( h ) is defined by ( h(x) = 4x + 28 ). the graph of ( y = h(x) ) in xy-plane has an x-intercept at ( (a, 0) ) and a y-intercept ( (0, b) ), where ( a ) and ( b ) are constants. what is the value of ( a + b ).
a. 21
b. 28
c. 32
d. 35

5. if the equation ( 5x - 12y = -30 ) is graphed in the xy-plane, what is the y-intercept?

6. if the equation ( y = -2x + 9 ) is graphed in the xy-plane, what is the x-intercept?

7. if the equation ( 8x + 5y - 70 = 10 ) is graphed in the xy-plane, what is the y-intercept?

8. if the equation ( y = \frac{5}{8}x - 15 ) is graphed in the xy-plane, what is the x-intercept?

Question

1. dana is riding in a 100-mile-long bicycle race from indy to cincinnati. the function ( r(x) = 100 - 18x ) models the number of miles that dana has remaining ( x ) hours after she starts the race. what is the best interpretation of ( r(2) = 64 )?
   a. dana rides about 64 miles in the first 2 hours of the race.
   b. dana will ride the last 64 miles of the race in about 2 hours.
   c. when dana has ridden 64 miles, she has about 2 hours remaining.
   d. two hours after dana starts the race, she has about 64 miles remaining.

2. a wading pool that already contains 5 gallons of water is being filled. the function ( w(x) = 2.1x + 5 ) models the amount of water in the pool after ( x ) minutes. what does ( w(5) = 15.5 ) mean in the context of the problem?

3. the graph of the equation ( y = \frac{2}{3}x + c ), where ( c ) is a constant intersects the x-axis at ( (12, 0) ). what is the value of ( c )?

4. the function ( h ) is defined by ( h(x) = 4x + 28 ). the graph of ( y = h(x) ) in xy-plane has an x-intercept at ( (a, 0) ) and a y-intercept ( (0, b) ), where ( a ) and ( b ) are constants. what is the value of ( a + b ).
   a. 21
   b. 28
   c. 32
   d. 35

5. if the equation ( 5x - 12y = -30 ) is graphed in the xy-plane, what is the y-intercept?

6. if the equation ( y = -2x + 9 ) is graphed in the xy-plane, what is the x-intercept?

7. if the equation ( 8x + 5y - 70 = 10 ) is graphed in the xy-plane, what is the y-intercept?

8. if the equation ( y = \frac{5}{8}x - 15 ) is graphed in the xy-plane, what is the x-intercept?

Accepted Answer

### Question 1
# Explanation:
## Step1: Understand the function
The function $ r(x) = 100 - 18x $ models the remaining miles $ x $ hours after starting. So $ x $ is time in hours, $ r(x) $ is remaining miles.

## Step2: Interpret $ r(2) = 64 $
Substitute $ x = 2 $ into the function. This means 2 hours after starting, we calculate remaining miles. $ r(2)=100 - 18\times2 = 100 - 36 = 64 $, so 2 hours after starting, remaining miles are 64.

# Answer:
D. Two hours after Dana starts the race, she has about 64 miles remaining.

### Question 2
# Explanation:
## Step1: Understand the function
The function $ w(x)=2.1x + 5 $ models water in gallons after $ x $ minutes. $ x $ is time in minutes, $ w(x) $ is water in gallons.

## Step2: Interpret $ w(5)=15.5 $
Substitute $ x = 5 $ into the function. So after 5 minutes, we calculate water amount. $ w(5)=2.1\times5 + 5=10.5 + 5 = 15.5 $, meaning after 5 minutes, the pool has 15.5 gallons.

# Answer:
After 5 minutes of filling, the wading pool contains 15.5 gallons of water.

### Question 3
# Explanation:
## Step1: Substitute the point into the equation
The line $ y=\frac{2}{3}x + c $ passes through $ (12,0) $. Substitute $ x = 12 $, $ y = 0 $ into the equation.

## Step2: Solve for $ c $
$ 0=\frac{2}{3}\times12 + c $
$ 0 = 8 + c $
Subtract 8 from both sides: $ c=-8 $

# Answer:
$ c = -8 $

### Question 4
# Explanation:
## Step1: Find x-intercept (a)
For $ h(x)=4x + 28 $, x-intercept is when $ y = 0 $ (i.e., $ h(x)=0 $).
$ 0 = 4x + 28 $
Subtract 28: $ -28 = 4x $
Divide by 4: $ x=-7 $, so $ a = -7 $

## Step2: Find y-intercept (b)
Y-intercept is when $ x = 0 $. $ h(0)=4\times0 + 28 = 28 $, so $ b = 28 $

## Step3: Calculate $ a + b $
$ a + b=-7 + 28 = 21 $? Wait, no, wait. Wait, the options have 21, 28, 32, 35. Wait, maybe I misread the x-intercept. Wait, the problem says "the graph of $ y = h(x) $ in xy-plane has an x-intercept at $ (a,0) $ and a y-intercept $ (0,b) $". Wait, let's recheck:

Wait, $ h(x)=4x + 28 $. X-intercept: set $ y = 0 $, $ 0 = 4x + 28 $ → $ x = -7 $, so $ a=-7 $. Y-intercept: set $ x = 0 $, $ y = 28 $, so $ b = 28 $. Then $ a + b=-7 + 28 = 21 $? But option A is 21. Wait, but maybe the x-intercept is positive? Wait, maybe the function is different? Wait, no, the function is $ h(x)=4x + 28 $. Wait, maybe I made a mistake. Wait, let's check again.

Wait, x-intercept: $ y = 0 $, so $ 4x + 28 = 0 $ → $ x = -7 $, so $ a = -7 $. Y-intercept: $ x = 0 $, $ y = 28 $, so $ b = 28 $. Then $ a + b = -7 + 28 = 21 $, which is option A.

# Answer:
A. 21

### Question 5
# Explanation:
## Step1: Rewrite in slope-intercept form ($ y = mx + b $)
Given $ 5x - 12y = -30 $. Solve for $ y $:
$ -12y = -5x - 30 $
Divide by -12: $ y=\frac{5}{12}x+\frac{30}{12}=\frac{5}{12}x + \frac{5}{2} $

## Step2: Identify y-intercept
In $ y = mx + b $, $ b $ is y-intercept. So $ b=\frac{5}{2}=2.5 $ or $ \frac{5}{2} $

# Answer:
The y-intercept is $ \frac{5}{2} $ (or 2.5).

### Question 6
# Explanation:
## Step1: Find x-intercept (set $ y = 0 $)
For $ y=-2x + 9 $, set $ y = 0 $:
$ 0=-2x + 9 $
Add $ 2x $: $ 2x = 9 $
Divide by 2: $ x=\frac{9}{2}=4.5 $

# Answer:
The x-intercept is $ \frac{9}{2} $ (or 4.5).

### Question 7
# Explanation:
## Step1: Rewrite the equation
Given $ 8x + 5y - 70 = 10 $. Simplify: $ 8x + 5y = 80 $

## Step2: Solve for y-intercept (set $ x = 0 $)
Set $ x = 0 $: $ 0 + 5y = 80 $
Divide by 5: $ y = 16 $

# Answer:
The y-intercept is 16.

### Question 8
# Explanation:
## Step1: Find x-intercept (set $ y = 0 $)
For $ y=\frac{5}{8}x - 15 $, set $ y = 0 $:
$ 0=\frac{5}{8}x - 15 $
Add 15: $ \frac{5}{8}x = 15 $
Multiply by $ \frac{8}{5} $: $ x = 15\times\frac{8}{5}=24 $

# Answer:
The x-intercept is 24.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1

Step1: Understand the function

Step2: Interpret \( r(2) = 64 \)

Step1: Understand the function

Step2: Interpret \( w(5)=15.5 \)

Step1: Substitute the point into the equation

Step2: Solve for \( c \)

Answer:

Question 2