Question

use your understanding of the constant of proportionality to answer the questions below. while training for a marathon, keiths watch reported the number of calories he had burned at each mile - marker. the data is shown below.

# of miles	1	2	3	4	5
# of calories burned	117	234	351	468	585

a. is the number of calories proportional to the number of miles? justify your thinking.

b. what is the constant of proportionality that relates y, the number of calories burned, to x, the number of miles?

c. write an equation to represent the relationship between the number of miles and the calories burned.

d. the next day, keith ran 7 miles. how many calories did he burn?

e. if keiths watch reported that he burned 1,170 calories, how many miles did keith run that day?

summarize todays lesson:

Explanation:

Step1: Check proportionality

To check if the number of calories $y$ is proportional to the number of miles $x$, we calculate the ratio $\frac{y}{x}$ for each pair of values.
For $x = 1,y=117$, $\frac{y}{x}=\frac{117}{1}=117$; for $x = 2,y = 234$, $\frac{y}{x}=\frac{234}{2}=117$; for $x = 3,y=351$, $\frac{y}{x}=\frac{351}{3}=117$; for $x = 4,y = 468$, $\frac{y}{x}=\frac{468}{4}=117$; for $x = 5,y=585$, $\frac{y}{x}=\frac{585}{5}=117$. Since the ratio $\frac{y}{x}$ is constant ($117$), the number of calories is proportional to the number of miles.

Step2: Find constant of proportionality

The constant of proportionality $k$ is the ratio of the number of calories burned to the number of miles. As calculated above, $k = 117$.

Step3: Write the equation

The equation for a proportional - relationship is $y=kx$. Substituting $k = 117$, we get $y = 117x$.

Step4: Calculate calories for 7 miles

Substitute $x = 7$ into the equation $y=117x$. Then $y=117\times7 = 819$.

Step5: Calculate miles for 1170 calories

Substitute $y = 1170$ into the equation $y = 117x$. Solve for $x$: $x=\frac{y}{117}=\frac{1170}{117}=10$.

Answer:

a. Yes, because the ratio of the number of calories burned to the number of miles is always 117.
b. 117
c. $y = 117x$
d. 819
e. 10