Question

for each linear relationship, determine the rate of change and interpret its meaning in the given context.

linear relationship	rate of change	interpretation
8. the cost of a large cheese pizza is $12.50 and each additional topping costs $0.75.
9. we left for our vacation at 11:00am the table indicates our mileage each hour during our trip. <table><thead><tr><th>time</th><th>mileage</th></tr></thead><tbody><tr><td>11:00</td><td>13,423</td></tr><tr><td>12:00</td><td>13,498</td></tr><tr><td>1:00</td><td>13,573</td></tr><tr><td>2:00</td><td>13,648</td></tr></tbody></table>
10. nathan is planning a birthday party. the cost for 8 people is $123 and the cost for 12 people is $167.

Explanation:

Problem 7

Step1: Identify two points

We have points \((0, 357)\) and \((2, 825)\) (assuming the first point is \((0, 357)\) from the graph's intercept and \((2, 825)\) as given).

Step2: Calculate rate of change (slope)

The formula for rate of change (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 \(x_1 = 0,y_1=357,x_2 = 2,y_2 = 825\), we get \(m=\frac{825 - 357}{2-0}=\frac{468}{2} = 234\).

Step3: Interpret the rate of change

The rate of change is \(234\) dollars per week. This means that Sarah is saving \$234 per week in her savings account.

Step1: Identify the linear relationship

The cost \(C\) of a pizza with \(t\) toppings is given by \(C=12.50 + 0.75t\).

Step2: Determine the rate of change

In the linear equation \(y=mx + b\), the rate of change is \(m\). Here, \(m = 0.75\).

Step3: Interpret the rate of change

The rate of change is \$0.75 per topping. This means that for each additional topping added to the large cheese pizza, the cost increases by \$0.75.

Step1: Identify two points

Take two consecutive points, e.g., \((11:00,13423)\) and \((12:00,13498)\). Let \(x\) be time (in hours after 11:00) and \(y\) be mileage. So \(x_1 = 0,y_1 = 13423,x_2=1,y_2 = 13498\).

Step2: Calculate rate of change (slope)

Using \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{13498 - 13423}{1-0}=\frac{75}{1}=75\).

Step3: Interpret the rate of change

The rate of change is \(75\) miles per hour. This means that the vehicle is traveling at a speed of 75 miles per hour during the trip.

Answer:

Rate of Change: \(234\) (dollars per week)
Interpretation: Sarah saves \$234 per week in her savings account.

Problem 8