Lesson 2 Homework Practice Volume of Cones Answer Key
The lesson 2 homework practice volume of cones answer key offers students a concise yet thorough guide to mastering the calculation of cone volumes. In this article you will find a clear explanation of the underlying formula, a step‑by‑step method for solving typical homework problems, common errors to watch out for, and a ready‑to‑use answer key with worked examples. By following these instructions, learners can confidently tackle any volume‑of‑a‑cone question and reinforce their understanding of three‑dimensional geometry And it works..
Understanding the Volume of a Cone
The Basic Formula
The volume (V) of a cone is determined by the formula
[ V = \frac{1}{3}\pi r^{2}h ]
where (r) is the radius of the circular base and (h) is the height measured from the base to the apex perpendicular to the base.
- ( \pi ) represents the constant ratio of a circle’s circumference to its diameter (approximately 3.14159).
- The factor ( \frac{1}{3} ) arises because a cone is essentially one‑third of a cylinder with the same base and height.
Key Terms
- Radius – the distance from the center of the base to its edge.
- Height – the vertical distance from the base to the tip of the cone.
Italic emphasis is used here for these technical terms to highlight their importance.
Step‑by‑Step Guide to Solving Homework Problems
Step 1: Identify the Given Values
- Read the problem carefully and note the dimensions provided.
- Determine whether the radius or diameter is given; if the diameter is known, divide it by 2 to obtain the radius.
Step 2: Choose the Correct Formula
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If the problem asks for the volume, use (V = \frac{1}{3}\pi r^{2}h).
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If the problem provides the volume and asks for a missing dimension (radius or height), rearrange the formula accordingly:
- For radius: (r = \sqrt{\frac{3V}{\pi h}})
- For height: (h = \frac{3V}{\pi r^{2}})
Step 3: Substitute Values
- Plug the known values into the formula, keeping units consistent (e.g., centimeters, meters).
- Bold the substituted expression to make it stand out, for example: (V = \frac{1}{3}\pi (4\text{ cm})^{2}(9\text{ cm})).
Step 4: Perform the Calculations
- Square the radius (if not already squared).
- Multiply by the height.
- Multiply by (\pi).
- Finally, multiply by (\frac{1}{3}).
Tip: Use a calculator for precision, but keep intermediate steps visible to avoid errors.
Step 5: State the Final Answer with Units
- Write the result as “(V = \text{value} \text{ cubic units})”.
- Ensure the unit matches the input units (e.g., cm³, m³).
Common Mistakes and How to Avoid Them
- Using diameter instead of radius: Always convert diameter to radius before squaring.
- Forgetting the (\frac{1}{3}) factor: Remember that a cone’s volume is one‑third of the corresponding cylinder.
- Mixing up units: Convert all measurements to the same unit system before calculation.
- Rounding too early: Keep full precision until the final answer, then round appropriately.
Answer Key Overview
Below is a comprehensive answer key featuring typical homework questions, the solution steps, and the final results. Each problem is labeled for easy reference Not complicated — just consistent..
Problem 1
Find the volume of a cone with a radius of 5 cm and a height of 12 cm.
Solution:
- Substitute into the formula: (V = \frac{1}{3}\pi (5\text{ cm})^{2}(12\text{ cm})).
- Calculate (5^{2} = 25).
- Multiply: (25 \times 12 = 300).
- Multiply by (\pi): (300\pi).
- Multiply by (\frac{1}{3}): (V = 100\pi) cm³.
Answer: (100\pi) cm³ (approximately 314.16 cm³) Practical, not theoretical..
Problem 2
A cone has a volume of 200 π cm³ and a height of 9 cm. Determine the radius of its base.
Solution:
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Rearrange for radius: (r = \sqrt{\frac{3V}{\pi h}}).
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Substitute (V = 200\pi) and (h = 9):
[ r = \sqrt{\frac{3 \times 200\pi}{\pi \times 9}} = \sqrt{\frac{600}{9}} = \sqrt{66.\overline{6}} \approx 8.13\text{ cm} ]
Answer: (r \approx 8.13) cm Small thing, real impact..
Problem 3
If the radius of a cone is 7 cm and its volume is 1,155 cm³, find the height.
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Problem 3
If the radius of a cone is 7 cm and its volume is 1,155 cm³, find the height.
Solution
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Rearrange the volume formula to solve for height
[ V = \frac{1}{3}\pi r^{2}h ;;\Longrightarrow;; h = \frac{3V}{\pi r^{2}} ] -
Substitute the known values (keep the units consistent):
[ \textbf{(h = \frac{3 \times 1155\text{ cm}³}{\pi (7\text{ cm})²})} ] -
Perform the calculations
- Square the radius: ((7\text{ cm})² = 49\text{ cm}²)
- Multiply the numerator: (
