Question

1. a football team played four quarters of a game. their total points scored after each quarter is shown.

quarter 1 2 3 4
score 7 14 14 21
a is the total points scored a function of the number of quarters? circle one.
yes no
explain your thinking.
b is there a linear relationship between the number of quarters and the total points scored? circle one.
yes no
explain your thinking.

1. each row of the table below represents a cone with a radius of 6 inches and a different height.

a calculate the volume of each cone. write your answer in terms of $pi$ or rounded to the nearest cubic inch.
height (in.) volume (cu. in.)
1
2
3
b is there a linear relationship between the height and the volume of these cones? circle one.
yes no
explain your thinking.

Explanation:

Problem 5

Part a

A function is defined such that each input (number of quarters) has exactly one output (total points). For each quarter number (1, 2, 3, 4), there is only one corresponding total score.

Part b

A linear relationship requires a constant rate of change (constant slope) between the two variables. Calculate the change in points per quarter:

• From Q1 to Q2: $14-7=7$ points added
• From Q2 to Q3: $14-14=0$ points added
• From Q3 to Q4: $21-14=7$ points added

The rate of change is not constant, so there is no linear relationship.

The formula for the volume of a cone is $V=\frac{1}{3}\pi r^2 h$, where $r=6$ inches. First, substitute $r=6$ into the formula:

Step1: Simplify the fixed terms

$V=\frac{1}{3}\pi (6)^2 h = \frac{1}{3}\pi (36)h = 12\pi h$

Step2: Calculate volume for h=1

Substitute $h=1$:
$V=12\pi(1)=12\pi \approx 38$ cu. in.

Step3: Calculate volume for h=2

Substitute $h=2$:
$V=12\pi(2)=24\pi \approx 75$ cu. in.

Step4: Calculate volume for h=3

Substitute $h=3$:
$V=12\pi(3)=36\pi \approx 113$ cu. in.

Step5: Check linear relationship

The simplified volume formula is $V=12\pi h$, which follows the linear form $y=mx$ (a proportional, linear relationship). The rate of change of volume with respect to height is constant ($12\pi$).

Answer:

a. Yes
b. No

---

Problem 6