Question

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lo: swbat solve problems involving the volume of cylinders, cones, and spheres.
application problems

1. simones mom is filling twelve birthday hats with candy for her birthday party. each hat is shaped like a cone with a height of 7 inches and a radius of 4 inches. which statements are true? choose two correct answers.

a. if the radius of each hat were 1 inch smaller and the height 1 inch greater, the hat would hold the same amount of candy.
b. the height of a hat does not affect how much candy the hat holds.
c. each hat can hold about 117.29 cubic inches of candy.
d. it will require more than 700 cubic inches of candy to fill all of the hats.
e. a hat with a radius of 4 inches and a height of 5 inches would hold more candy than the hats simones mom is using.

1. beth has a can of corn that is 3.5 inches high and has a diameter of 5 inches. find the volume of the can to the nearest tenth.

a. 192.4 in.³
b. 68.7 in.³
c. 274.9 in.³
d. 55.0 in.³

1. chris has a cone with a radius of 5 inches and a height of 12 inches. what is the approximate volume of the cone?

a. 377.00 in.³
b. 188.50 in.³
c. 942.48 in.³
d. 314.16 in.³

1. cody uses a battery shaped like a cylinder for his flashlight. the radius of the battery is 16.5 mm, and it has a volume of 15,908.5 cubic mm. how tall is the battery?

a. 56.3 mm
b. 16.5 mm
c. 18.6 mm
d. 59.3 mm

Explanation:

Step1: Calculate the volume of the cone

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\).
Given \(r = 4\) inches and \(h=7\) inches.
Substitute into the formula:
\(V=\frac{1}{3}\times\pi\times4^{2}\times7=\frac{1}{3}\times\pi\times16\times7=\frac{112\pi}{3}\approx117.29\) cubic inches.

Step2: Analyze option C

Since \(V\approx117.29\) cubic inches, option C is correct.

Step3: Calculate the total volume of 12 hats

The total volume \(V_{total}=12\times V = 12\times117.29 = 1407.48\) cubic inches.
Since \(1407.48>700\), option D is correct.

Step4: Analyze option A

Original volume \(V_1=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times4^{2}\times7\).
New radius \(r'=4 - 1=3\) inches, new height \(h'=7 + 1 = 8\) inches.
New volume \(V_2=\frac{1}{3}\pi\times3^{2}\times8=\frac{1}{3}\pi\times9\times8 = 24\pi\).
\(V_1=\frac{112\pi}{3}\approx117.29\), \(V_2 = 24\pi\approx75.4\). So option A is wrong.

Step5: Analyze option B

From the formula \(V=\frac{1}{3}\pi r^{2}h\), height \(h\) affects the volume, so option B is wrong.

Step6: Analyze option E

Original volume \(V=\frac{1}{3}\pi\times4^{2}\times7\approx117.29\).
New volume \(V'=\frac{1}{3}\pi\times4^{2}\times5=\frac{80\pi}{3}\approx83.78\). So option E is wrong.

Answer:

C. Each hat can hold about 117.29 cubic inches of candy.
D. It will require more than 700 cubic inches of candy to fill all of the hats.