. The tent instructions reveal the height at the tallest point to be 4.5 feet and the space inside to be 10.5 cubic feet. About how many square feet of the back porch will be covered by the tent? Which measure of the cone needs to be calculated to answer the problem? A. diameter B. Radius C. Base area D. Volume. Which equation represents the scenario? A. V = (1/3)(10.5)(4.5) B. 4.5 = (1/3)(10.5)(h) C. 10.5 = (1/3)(4.5)(h) D. 10.5 = (1/3)(B)(4.5). How many square feet of the back porch will the tent cover? A. 2.33 square feet B. 6 square feet C. 7 square feet D. 15.75 square feet.
The Correct Answer and Explanation is:
We use the volume formula for a cone:V=13πr2hV = \frac{1}{3} \pi r^2 hV=31πr2h
We are given V=10.5V = 10.5V=10.5 and h=4.5h = 4.5h=4.5, and we need to solve for rrr, which helps us find the base area.
Let’s substitute the known values into the formula:10.5=13πr2(4.5)10.5 = \frac{1}{3} \pi r^2 (4.5)10.5=31πr2(4.5)
Simplify:10.5=4.53πr2=1.5πr210.5 = \frac{4.5}{3} \pi r^2 = 1.5 \pi r^210.5=34.5πr2=1.5πr2
Now divide both sides by 1.5π1.5\pi1.5π:r2=10.51.5π=10.54.71238898≈2.23r^2 = \frac{10.5}{1.5\pi} = \frac{10.5}{4.71238898} \approx 2.23r2=1.5π10.5=4.7123889810.5≈2.23
Then:r≈2.23≈1.49 feetr \approx \sqrt{2.23} \approx 1.49 \text{ feet}r≈2.23≈1.49 feet
Now we find the base area using:A=πr2≈π(1.49)2≈3.14×2.23≈7.0 square feetA = \pi r^2 \approx \pi (1.49)^2 \approx 3.14 \times 2.23 \approx 7.0 \text{ square feet}A=πr2≈π(1.49)2≈3.14×2.23≈7.0 square feet
So, the area of the base, or how much of the back porch the tent covers, is about 7 square feet.
Correct Answers:
- Which measure? → C. Base area
- Which equation? → D. 10.5=13B⋅4.510.5 = \frac{1}{3} B \cdot 4.510.5=31B⋅4.5
- Area of porch covered? → C. 7 square feet
Explanation
To find how much space the conical tent covers on the porch, we need to determine the area of the base of the cone, since that is the part in contact with the porch. A cone’s base is a circle, and the area of a circle is given by the formula A=πr2A = \pi r^2A=πr2, where rrr is the radius.
However, we are not given the radius directly. Instead, we are given the volume of the cone (10.5 cubic feet) and the height (4.5 feet). We use the formula for the volume of a cone to find the missing radius:V=13πr2hV = \frac{1}{3} \pi r^2 hV=31πr2h
Substituting the known values into the formula, we get:10.5=13πr2⋅4.510.5 = \frac{1}{3} \pi r^2 \cdot 4.510.5=31πr2⋅4.510.5=1.5πr210.5 = 1.5 \pi r^210.5=1.5πr2
Now solve for r2r^2r2:r2=10.51.5π≈10.54.71≈2.23r^2 = \frac{10.5}{1.5\pi} \approx \frac{10.5}{4.71} \approx 2.23r2=1.5π10.5≈4.7110.5≈2.23r≈2.23≈1.49 feetr \approx \sqrt{2.23} \approx 1.49 \text{ feet}r≈2.23≈1.49 feet
Finally, we plug the radius back into the area formula:A=πr2≈3.14×2.23≈7.0A = \pi r^2 \approx 3.14 \times 2.23 \approx 7.0A=πr2≈3.14×2.23≈7.0
So, the tent covers about 7 square feet of the porch. Understanding how to apply geometric formulas and solve for unknown variables is key when working with shapes like cones in real-world contexts.
