At the bowling alley, Terry paid $22.50 and bowled 3 games while Tommy bowled 2 games and paid $15. Interpret the rate of change by identifying the cost of each game.(1 point)

Question

Use the image to answer the question.
An illustration shows three coiled wire springs stretched to varying lengths. The first spring does not have any weights pulling it downward, and the length of the spring is marked as a question mark inches. The second spring has a 2-pound weight attached to the bottom, and its length is marked as 12 inches. The third spring has a 5-pound weight attached to the bottom and its length is marked as 18 inches.

A spring has a length of 12 inches when a 2-pound weight is attached, and a length of 18 inches when a 5-pound weight is attached. Use rate of change to find the length of the spring when no weights are attached.

The Kims are hosting a catered dinner. The cost for 3 servings is $18. The cost for 10 servings is $60. What is the cost per serving?(1 point)

A graph of the cost of gas purchased depending on the number of gallons pumped has the points (4,15)
and (8,30)
. What is the rate of change in the cost of the gas as each gallon is pumped?

A linear graph of the cost of broccoli in dollars per pound has the points (3,8.25)
and (6,16.50)
. What is the rate of change in the purchase price in dollars per pound?

Answer 1

Let's break down each problem and find the required solutions.

Problem 1: Cost of Bowling Games

Terry paid $22.50 for 3 games: \[ \text{Cost per game for Terry} = \frac{\text{Total Cost}}{\text{Number of Games}} = \frac{22.50}{3} = 7.50 \]

Tommy paid $15 for 2 games: \[ \text{Cost per game for Tommy} = \frac{15}{2} = 7.50 \]

Thus, the cost of each game is $7.50.

Problem 2: Spring Length with No Weight

We know that:

• 2-pound weight → 12 inches
• 5-pound weight → 18 inches

To find the rate of change (length per pound), we can calculate:

• Change in length = 18 inches - 12 inches = 6 inches
• Change in weight = 5 pounds - 2 pounds = 3 pounds

Rate of change (length per pound) is: \[ \text{Rate of Change} = \frac{6 \text{ inches}}{3 \text{ pounds}} = 2 \text{ inches per pound} \]

Now, we can use the 2-pound weight to find the length of the spring with no weight: \[ \text{Length with no weight} = \text{Length with 2 pounds} - (\text{Weight} \times \text{Rate of Change}) = 12 \text{ inches} - (2 \text{ pounds} \times 2 \text{ inches/pound}) = 12 - 4 = 8 \text{ inches} \]

The length of the spring when no weights are attached is 8 inches.

Problem 3: Cost Per Serving for Kims' Dinner

For 3 servings ($18): \[ \text{Cost per serving} = \frac{18}{3} = 6 \]

For 10 servings ($60): \[ \text{Cost per serving} = \frac{60}{10} = 6 \]

Thus, the cost per serving is $6.

Problem 4: Rate of Change for Gas Cost

Given points (4,15) and (8,30): \[ \text{Rate of Change} = \frac{30 - 15}{8 - 4} = \frac{15}{4} = 3.75 \]

The rate of change in the cost of gas as each gallon is pumped is $3.75 per gallon.

Problem 5: Rate of Change for Broccoli Cost

Given points (3,8.25) and (6,16.50): \[ \text{Rate of Change} = \frac{16.50 - 8.25}{6 - 3} = \frac{8.25}{3} = 2.75 \]

The rate of change in the purchase price of broccoli is $2.75 per pound.

Summary of Answers:

1. Cost per game (bowling): $7.50
2. Length of spring with no weight: 8 inches
3. Cost per serving: $6
4. Rate of change in gas cost: $3.75 per gallon
5. Rate of change in broccoli cost: $2.75 per pound

Answer 2

Use the table to answer the question.

Height Above Canyon Floor Time (minutes)
360 0
280 3
160 10
80 14
Which statement best interprets the rate of change of the linear model shown in the table?

(1 point)
Responses

The elevation of a hiker who is hiking up from canyon floor changes at a rate of −20
feet per minute.
The elevation of a hiker who is hiking up from canyon floor changes at a rate of negative 20 feet per minute.

The elevation of a hiker who is hiking down to a canyon floor changes at a rate of 20 feet per minute.
The elevation of a hiker who is hiking down to a canyon floor changes at a rate of 20 feet per minute.

The elevation of a hiker who is hiking down to a canyon floor changes at a rate of −20
feet per minute.
The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute.

The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute.

Use the table to answer the question.

Number of Months Cost ($)
1 74
2 99
3 124
4 149
The table below shows the cost for a gym membership at the local fitness center with an initial start-up fee charged in the first month. Find the slope, or monthly cost.

(1 point)
Responses

$25
$25

$74
$74

$37.25
$37.25

$49

A linear graph of parking fees in dollars based on hours parked has the points (2,20)
and (6,44)
. How would you interpret the slope of the graph as the rate of change in the parking fee for each additional hour of parking?(1 point)
Responses

The parking fee rises by $10 with each additional hour.
The parking fee rises by $10 with each additional hour.

The parking fee rises by $7.33 with each additional hour.
The parking fee rises by $7.33 with each additional hour.

The parking fee rises by $6 with each additional hour.
The parking fee rises by $6 with each additional hour.

The parking fee rises by $8 with each additional hour.

Use the image to answer the question.
A coordinate plane shows the x-axis, labeled as hours, ranging from 0 to 6 in increments of 1 and the y-axis, labeled as gallons, ranging from 0 to 240 in increments of 20. A line segment joins four plotted points. The coordinates of the plotted points are left parenthesis 0 comma 200 right parenthesis, left parenthesis 1 comma 160 right parenthesis, left parenthesis 3 comma 80 right parenthesis, and left parenthesis 5 comma 0 right parenthesis.

The graph shows the number of gallons in a wading pool that is being drained. Using the points (1,160)
and (3,80)
as (x1,y1)
and (x2,y2)
, what is the rate of change in the number of gallons in the pool after each hour has passed?

(1 point)
Responses

1/40
gal./hr.
Start Fraction 1 over 40 End Fraction gal./hr.

−1/40
gal./hr.
negative Start Fraction 1 over 40 End Fraction gal./hr.

40 gal./hr.
40 gal./hr.

−40
gal./hr.
negative 40 gal./hr.

Question
Which formula can be used to find the slope of a line from two points (x1,y1)
and (x2,y2)
?(1 point)
Responses

y2/y1−x2/x1
Start Fraction y subscript 2 baseline over y subscript 1 baseline End Fraction minus Start Fraction x subscript 2 baseline over x subscript 1 baseline End Fraction

y2−y1/x2−x1
Start Fraction y subscript 2 baseline minus y subscript 1 baseline over x subscript 2 baseline minus x subscript 1 baseline End Fraction

y2+y1/x2+x1
Start Fraction y subscript 2 baseline plus y subscript 1 baseline over x subscript 2 baseline plus x subscript 1 baseline End Fraction

x2−x1/y2−y1
Start Fraction x subscript 2 baseline minus x subscript 1 baseline over y subscript 2 baseline minus y subscript 1 baseline End Fraction

Answer 3

Let's go through each question one by one.

1. Rate of Change Interpretation from Table

Given the height above the canyon floor and time data:

• Height decreases from 360 to 280 in 3 minutes.
• Height decreases from 280 to 160 in 7 minutes.
• Height decreases from 160 to 80 in 4 minutes.

Overall, the height changes from 360 to 80 over 14 minutes, which is a total change of: \[ 360 - 80 = 280 \text{ feet} \] The time for this change is 14 minutes.

Rate of Change: \[ \text{Rate of Change} = \frac{\text{Change in Height}}{\text{Change in Time}} = \frac{-280}{14} = -20 \text{ feet per minute} \]

The best interpretation is: The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute.

2. Monthly Cost Interpretation from Table

To find the slope (monthly cost) from the gym membership data:

• Month 1: $74
• Month 2: $99 → Increase: $99 - $74 = $25
• Month 3: $124 → Increase: $124 - $99 = $25
• Month 4: $149 → Increase: $149 - $124 = $25

The monthly cost is consistent at: $25.

3. Parking Fees Rate of Change Interpretation

Given points (2,20) and (6,44): \[ \text{Rate of Change} = \frac{44 - 20}{6 - 2} = \frac{24}{4} = 6 \text{ dollars per hour} \] This does not match any of the provided options. We should check the fee for additional hours.

By analyzing how the parking fee changes with respect to the time intervals: From hour 2 to hour 6:

• $20 (at 2 hrs) to $44 (at 6 hrs), a net increase of $24 over 4 hours, gives us:
• Thus, it seems an error exists in the question, as the correct interpretation of the slope based on the points should state: The parking fee rises by $6 with each additional hour.

4. Rate of Change in Gallons for Wading Pool

Using points (1,160) and (3,80): \[ \text{Rate of Change} = \frac{80 - 160}{3 - 1} = \frac{-80}{2} = -40 \text{ gallons/hour} \] Thus, the correct answer is: −40 gallons/hour.

5. Formula for Finding the Slope of a Line

The formula for slope from two points (x1, y1) and (x2, y2) is: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] Therefore, the correct answer is: y2 − y1 / x2 − x1.

Summary of Answers:

1. The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute.
2. $25.
3. The parking fee rises by $6 with each additional hour.
4. −40 gallons/hour.
5. y2 − y1 / x2 − x1.