Question

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

1. sarah started a savings account, the graph represents the relationship between the amount of money in the savings account, s, and the number of weeks w.

1. the cost of a large cheese pizza is $12.50 and each additional topping costs $0.75.

1. we left for our vacation at 11:00am the table indicates our mileage each hour during our trip.

1. nathan is planning a birthday party. the cost for 8 people is $123 and the cost for 12 people is $167.

Explanation:

Step1: Calculate rate of change for problem 7

The rate of change (slope) formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the points \((0,375)\) and \((2,1005)\) in problem 7, \(x_1 = 0,y_1=375,x_2 = 2,y_2 = 1005\). Then \(m=\frac{1005 - 375}{2-0}=\frac{630}{2}=315\).

Step2: Interpret rate of change for problem 7

The rate of change \(m = 315\) means that the amount of money in Sarah's savings account increases by \(315\) dollars per week.

Step3: Calculate rate of change for problem 8

The cost function is \(C(t)=12.50+0.75t\), where \(t\) is the number of additional toppings. The rate of change (slope) of the linear function \(y = mx + b\) is \(m\). Here, \(m = 0.75\).

Step4: Interpret rate of change for problem 8

The rate of change \(m=0.75\) means that the cost of the pizza increases by \(0.75\) dollars for each additional topping.

Step5: Calculate rate of change for problem 9

Using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the points \((11,13423)\) and \((12,13498)\), \(m=\frac{13498 - 13423}{12 - 11}=75\). (We can check with other consecutive points: \(\frac{13573-13498}{13 - 12}=75\), \(\frac{13648 - 13573}{14 - 13}=75\))

Step6: Interpret rate of change for problem 9

The rate of change \(m = 75\) means that the mileage increases by \(75\) miles per hour.

Step7: Calculate rate of change for problem 10

Let \(x\) be the number of people and \(y\) be the cost. For the points \((8,123)\) and \((12,167)\), \(m=\frac{167-123}{12 - 8}=\frac{44}{4}=11\)

Step8: Interpret rate of change for problem 10

The rate of change \(m = 11\) means that the cost of the birthday party increases by \(11\) dollars per person.

Answer:

Linear Relationship	Rate of Change	Interpretation
8. Cost of pizza	\(0.75\)	The cost of the pizza increases by \(0.75\) dollars per additional topping.
9. Vacation mileage	\(75\)	The mileage increases by \(75\) miles per hour.
10. Birthday - party cost	\(11\)	The cost of the party increases by \(11\) dollars per person.