Jessica built a cone for science class. The volume of the cone was 42 cubic inches. The radius of the cone was 2 inches. What was the approximate height of the cone? A 40 inches a. 20 inches c. 10 inches D. 4 inches
The Correct Answer and Explanation is:
To find the height of a cone, we use the formula for the volume of a cone: V=13πr2hV = \frac{1}{3} \pi r^2 h
Where:
- VV is the volume,
- rr is the radius, and
- hh is the height.
We are given:
- Volume V=42V = 42 cubic inches
- Radius r=2r = 2 inches
We need to solve for hh, the height.
Step-by-step Calculation:
Start with the formula: 42=13π(2)2h42 = \frac{1}{3} \pi (2)^2 h 42=13π(4)h42 = \frac{1}{3} \pi (4) h 42=43πh42 = \frac{4}{3} \pi h
Now divide both sides by 43π\frac{4}{3} \pi: h=4243πh = \frac{42}{\frac{4}{3} \pi} h=42×34πh = \frac{42 \times 3}{4 \pi} h=1264πh = \frac{126}{4\pi}
Now approximate π≈3.14\pi \approx 3.14: h≈1264×3.14h \approx \frac{126}{4 \times 3.14} h≈12612.56≈10.03h \approx \frac{126}{12.56} \approx 10.03
So, the approximate height is 10 inches.
Final Answer:
C. 10 inches
Explanation
To solve this problem, we apply the volume formula of a cone, which is one-third the product of the base area and the height. A cone’s base is circular, so the area of the base is πr2\pi r^2. When we plug this into the volume formula, it becomes V=13πr2hV = \frac{1}{3} \pi r^2 h.
In this case, we know the volume is 42 cubic inches and the radius is 2 inches. Plugging these values into the formula lets us isolate and solve for the height hh. We square the radius (2 inches) to get 4. Multiply by π\pi and then by 13\frac{1}{3}, giving 43π\frac{4}{3} \pi. We set this equal to the known volume (42) and solve algebraically to isolate hh.
After rearranging, we divide 42 by 43π\frac{4}{3} \pi, which simplifies to 1264π\frac{126}{4 \pi}. Using the approximation π≈3.14\pi \approx 3.14, we compute 4×3.14=12.564 \times 3.14 = 12.56, then divide 126 by 12.56. This results in approximately 10.03 inches.
The closest whole number is 10, so the best approximation of the cone’s height is 10 inches. This matches option C, making it the correct answer. This process shows how mathematical formulas, substitutions, and approximations work together to solve real-world geometry problems.
