Question

finding an unknown dimension4. at a candle factory, a glass cone-shaped jar with a volume of 1,206.4 in³ holds melted candle wax. the jar is inside an 8-in.-tall cylinder that has the same radius and height as the jar. how much space is between the wall of the cylinder and the wall of the cone?5. the parents of the students on the cheer squad found some 24-in. cardboard cones at the craft store and decided to cut 3 in. off the tips of the cones to make them look like megaphones for their centerpieces. they will fill each megaphone with beads in the school colors. the volume of one cone before the end is cut off is 2,034.72 in³.a. what is the diameter of the circle where the end is cut off the cone? (hint: use similar triangles.)b. what is the volume of the part of the cone the parents will use as the megaphone?

Explanation:

Step1: Find base radius of full cone

First, use the cone volume formula $V=\frac{1}{3}\pi r^2 h$. We know $V=2034.72\ \text{in}^3$, $h=24\ \text{in}$, $\pi=3.14$.
Rearrange to solve for $r$:

$$ r^2 = \frac{3V}{\pi h} = \frac{3\times2034.72}{3.14\times24} $$

$$ r^2 = \frac{6104.16}{75.36} = 81 $$

$$ r = \sqrt{81} = 9\ \text{in} $$

Step2: Use similar triangles for part (a)

The small cut-off cone has height $h_s=3\ \text{in}$. Let its radius be $r_s$. Similar triangles give $\frac{r_s}{r}=\frac{h_s}{h}$.

$$ r_s = r\times\frac{h_s}{h} = 9\times\frac{3}{24} = 1.125\ \text{in} $$

Diameter $d=2r_s$:

$$ d=2\times1.125=2.25\ \text{in} $$

Step3: Volume of small cone (part b prep)

Use cone volume formula for the small cut-off cone:

$$ V_s = \frac{1}{3}\pi r_s^2 h_s = \frac{1}{3}\times3.14\times(1.125)^2\times3 $$

$$ V_s = 3.14\times1.265625 = 3.9740625\ \text{in}^3 $$

Step4: Volume of megaphone (part b)

Subtract small cone volume from full cone volume:

$$ V_{\text{megaphone}} = V - V_s = 2034.72 - 3.9740625 $$

Answer:

a. $2.25$ inches
b. $2030.7459375$ cubic inches