Unit 5 Functions And Volume — Unit Plan

In each row, the output should be one more than half of the input.

1. Yes. Sample response: The number of seconds to wait depends on the number of minutes to wait.

   

2. No, if I know how many minutes I have to wait in line, I do not necessarily know how many people are in line. Sample response: The number of people who have to wait cannot be determined by the amount of time someone has to wait. For example, there could be 50 people waiting, or there could be 100 people waiting.

1. See diagram:

   

2. $v = 25n$.  This reflects the statement that the value (in cents) of my collection of quarters is always 25 times the number of quarters I have.

3. When the input is 10, the output is 250 (since $250=25\boldcdot  10$).

4. $n$ is the independent variable, and $v$ is the dependent variable.

1. 20. See graph in part 2.

2. 16

   

3. After taking 7 rides, there will be $27.50 remaining on the card.

1. 10 kilometers per hour
2. 3 kilometers into the race
3. Sample response: From 5 kilometers to 6 kilometers, Diego went faster, but he slowed down from 6 kilometers to 8 kilometers. He sped up again from 8 kilometers to 9 kilometers and finished the last kilometer at the same speed.

The first graph most directly reflects Elena’s story if the vertical axis represents Elena’s distance from home and the horizontal axis represents the time since she started to walk home from school the first time. The graph then demonstrates that the distance from home is a function of the time elapsed.

Less than. From the table, we see that the area of a square of side length 2 inches is 4 square inches, whereas from the equation, we find that the area of a circle with radius 1.2 inches is about 4.52 square inches.

1. D
2. B
3. Graph A does not make sense because there is a constant amount of daylight. Graph C does not make sense because it goes through the origin, meaning it started with 0 minutes of daylight.

Predictions between 20 and 22 thousand sales, depending on the data points used for the model, are reasonable.

Sample response: Yes. After 48 months, they sold about 20.5 thousand games. From Month 12 to Month 36, the rate of games sold was about $\frac{11}{24}$ thousand games per month. This means the amount sold during the 12 months from Month 36 to Month 48 was 5.5 thousand, since $\frac{11} {24} \boldcdot 12=5.5$, and 5.5 thousand added to 15 thousand is 20.5 thousand.

1. Lin started using her phone 2 hours after noon, or at 2:00 p.m., since that is where the negative slope begins.
2. Lin started charging her phone 8 hours after noon, or at 8:00 p.m., since that is where the positive slope begins.
3. The battery was dying at 30% per hour since it decreased 60% over 2 hours.

Cylinder $b$. Sample reasoning: A cylinder with a large radius would have a smaller change in height (slope) for the same volume of water added when compared to a cylinder with a smaller radius. Since the line for $b$ has the smaller slope, it must be the cylinder with the larger radius.

Sample response: About 11 cups of rice since it should be a little more than the box.

1. $16\pi$ cm². The square of the radius of the base is $4^2=16$, which is multiplied by $\pi$, giving $\pi\boldcdot 4^2=16\pi$.
2. $112\pi$ cm³. The height of the cylinder is 7, which is multiplied by the area of the base, giving $16\pi\boldcdot 7=112\pi$.
3. 351.68 cm³, because $112\boldcdot 3.14 \approx 351.68$

The radius is 2 inches, and the height is 3 inches. Since the diameter is 4 inches, the radius is half of 4 inches. The volume is $12\pi=2^2\pi h$, which means $12\pi=4\pi h$ and $h=3$.

Cylinder: $36\pi$ cubic units, because $\pi \boldcdot 3^2 \boldcdot 4 =36\pi$

Cone: $36\pi$ cubic units, because $\frac13 \pi \boldcdot 3^2 \boldcdot 12 = 36\pi$

Sample responses:

• Height and radius both 3 inches since $\frac{1}{3} \pi \boldcdot 3^2 \boldcdot 3 = 9\pi$.
• Radius 2 inches and height 6.75 inches since $\frac{1}{3} \pi \boldcdot 2^2 \boldcdot 6.75 = 9\pi$.
• Radius 1 inch and height 27 inches since $\frac{1}{3} \pi \boldcdot 1^2 \boldcdot 27 = 9\pi$.
• Radius 9 inches and height $\frac{1}{3}$ inches since $\frac{1}{3} \pi \boldcdot 9^2 \boldcdot \frac{1}{3} = 9\pi$. (This cone may look more like a plate, but it solves the problem.)

1. Sample response: The point $(10, 31.4)$ is on the line. A cylinder with radius 1ft and height 10 ft will have a volume of 31.4 ft³.
2. Sample response: I know this relationship is a function because the equation is in the form $y = mx+b$ and all linear relationships are functions.

1. $V = \pi c^3$
2. If the value of $c$ is halved, then the value of the volume would be $\frac18$ of the original volume since $\pi \left(\frac12 c\right)^3=\pi c^3 \left(\frac12\right)^3=\frac18 \pi c^3$.

Sample responses:

• Less than 500 cubic inches. The volume of the box that the hemisphere fits in is $10^2\boldcdot 5$, and the hemisphere does not take up all the space in the box.
• Less than $125\pi$ cubic inches. The volume of the cylinder that the hemisphere fits in is $\pi (5)^2 \boldcdot 5$, and the hemisphere does not take up all the space in the cylinder.
• More than $\frac{125}3\pi$ cubic inches. The volume of the cone that fits in the hemisphere is $\frac13 \pi (5)^2 \boldcdot 5$, and the hemisphere is larger than the cone.

1. $\frac{256}{3} \pi$ (or $85.33\pi$) cubic feet, because $V=\frac43 \pi (4)^3$
2. 33.49 cubic feet, because $V=\frac43 \pi (2)^3$

B, C, A, D

Sphere A has a radius of 4, so its volume is $\frac{256}3\pi$.

Sphere B has a diameter of 6, so its radius is 3, and its volume is $36\pi$.

Sphere C has a volume of 64$\pi$.

Sphere D has a radius twice as large as sphere B, so its radius is 6, and its volume is $288\pi$.