Use the table to answer the question.

Question

Height Above Canyon Floor:
360
280
160
80

Time (minutes)
0
3
10
14

Which statement best interprets the rate of change of the linear model shown in the table?

options:

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 up from a canyon floor changes at a rate of 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.

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.

Answer 1

To interpret the rate of change of the linear model based on the provided data, we need to calculate the rate of change in height over time.

Here’s the data in the table:

• At time 0 minutes, the height is 360 feet.
• At time 3 minutes, the height is 280 feet.
• At time 10 minutes, the height is 160 feet.
• At time 14 minutes, the height is 80 feet.

Now, let's calculate the overall change in height and time:

1. Total height change:

   • From 360 feet (at 0 minutes) to 80 feet (at 14 minutes):
   • Change in height = 80 - 360 = -280 feet

2. Total time change:

   • From 0 minutes to 14 minutes = 14 minutes

3. Rate of change:

   • Rate of change = Change in height / Change in time
   • Rate of change = -280 feet / 14 minutes = -20 feet per minute

The negative sign indicates that the hiker is descending.

Thus, the best interpretation of the rate of change from the options provided is:

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

Answer 2

Use the table to answer the question.

Number of Months:
1
2
3
4

Cost ($)
74
99
124
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.

Answer 3

To find the slope, or the monthly cost, from the given data, we need to analyze the change in cost over the change in time (number of months).

Here’s the data from the table:

• For 1 month: Cost = $74
• For 2 months: Cost = $99
• For 3 months: Cost = $124
• For 4 months: Cost = $149

Now, we can calculate the change in cost and the change in the number of months to determine the slope:

1. Calculating the change in cost and change in months:

   • From month 1 to month 2:

     • Change in cost: \( 99 - 74 = 25 \)
     • Change in months: \( 2 - 1 = 1 \)

   • From month 2 to month 3:

     • Change in cost: \( 124 - 99 = 25 \)
     • Change in months: \( 3 - 2 = 1 \)

   • From month 3 to month 4:

     • Change in cost: \( 149 - 124 = 25 \)
     • Change in months: \( 4 - 3 = 1 \)

2. Calculating the slope:

   Since the change in cost for every additional month is consistently $25, the slope (monthly cost) is:

   \[ \text{slope} = \frac{\text{Change in cost}}{\text{Change in months}} = \frac{25}{1} = 25 \]

Therefore, the monthly cost for the gym membership is $25.

Answer 4

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?

options:

The parking fee rises by $8 with each additional hour.
The parking fee rises by $8 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 $7.33 with each additional hour.
The parking fee rises by $7.33 with each additional hour.

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

Answer 5

To interpret the slope of the graph as the rate of change in the parking fee for each additional hour of parking, we need to calculate the slope using the two given points: (2, 20) and (6, 44).

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting in our points:

• \((x_1, y_1) = (2, 20)\)
• \((x_2, y_2) = (6, 44)\)

Now, substituting these values into the slope formula:

\[ m = \frac{44 - 20}{6 - 2} = \frac{24}{4} = 6 \]

So, the slope is \( 6 \), which means the rate of change in the parking fee is $6 for each additional hour of parking.

Thus, the correct interpretation from the options provided is:

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