Question

problem 4
complete this table.
the cylinder and cone in each row have the same dimensions.

	diameter (units)	radius (units)	base area (sq. units)	height (units)	cylinder volume (cu. units)	cone volume (cu. units)
row 2	6					40π
row 3			36π			48π

Answer:

To solve the problem, we analyze each row of the table using the formulas for the volume of a cylinder (\( V_{\text{cylinder}} = \text{Base Area} \times \text{Height} \)) and the volume of a cone (\( V_{\text{cone}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \)). We also use the relationship between radius and diameter (\( \text{Radius} = \frac{\text{Diameter}}{2} \)) and the formula for the area of a circle (\( \text{Base Area} = \pi r^2 \)).

First Row (Radius = 5, Height = 7)

1. Diameter: \( \text{Diameter} = 2 \times \text{Radius} = 2 \times 5 = 10 \)
2. Base Area: \( \text{Base Area} = \pi r^2 = \pi (5)^2 = 25\pi \)
3. Cylinder Volume: \( V_{\text{cylinder}} = \text{Base Area} \times \text{Height} = 25\pi \times 7 = 175\pi \)
4. Cone Volume: \( V_{\text{cone}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 25\pi \times 7 = \frac{175\pi}{3} \)

Second Row (Diameter = 6, Cone Volume = \( 40\pi \))

1. Radius: \( \text{Radius} = \frac{\text{Diameter}}{2} = \frac{6}{2} = 3 \)
2. Base Area: \( \text{Base Area} = \pi r^2 = \pi (3)^2 = 9\pi \)
3. Cylinder Volume: Since \( V_{\text{cylinder}} = 3 \times V_{\text{cone}} \) (for the same base and height), \( V_{\text{cylinder}} = 3 \times 40\pi = 120\pi \)
4. Height: Using \( V_{\text{cone}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \), solve for height:

\( 40\pi = \frac{1}{3} \times 9\pi \times \text{Height} \)
\( 40\pi = 3\pi \times \text{Height} \)
\( \text{Height} = \frac{40\pi}{3\pi} = \frac{40}{3} \) (Wait, but the table shows "40π" for cone volume. Wait, maybe a miscalculation. Wait, the table has "40π" as cone volume. Let's recheck:
\( V_{\text{cone}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \)
\( 40\pi = \frac{1}{3} \times 9\pi \times h \)
\( 40\pi = 3\pi h \)
\( h = \frac{40\pi}{3\pi} = \frac{40}{3} \approx 13.33 \). But maybe the table has a typo or I misread. Wait, the table shows "40π" for cone volume. Alternatively, maybe the base area is 36π? Wait, third row: Base Area = 36π, Cone Volume = 48π. Let's check third row.

Third Row (Base Area = \( 36\pi \), Cone Volume = \( 48\pi \))

1. Radius: \( \text{Base Area} = \pi r^2 = 36\pi \implies r^2 = 36 \implies r = 6 \)

Diameter: \( \text{Diameter} = 2 \times 6 = 12 \)

1. Height: Using \( V_{\text{cone}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \):

\( 48\pi = \frac{1}{3} \times 36\pi \times \text{Height} \)
\( 48\pi = 12\pi \times \text{Height} \)
\( \text{Height} = \frac{48\pi}{12\pi} = 4 \)

1. Cylinder Volume: \( V_{\text{cylinder}} = \text{Base Area} \times \text{Height} = 36\pi \times 4 = 144\pi \) (Wait, the table shows "48π" for cone volume. Wait, maybe I made a mistake. Wait, \( V_{\text{cylinder}} = 3 \times V_{\text{cone}} \) when base and height are the same. So if \( V_{\text{cone}} = 48\pi \), then \( V_{\text{cylinder}} = 3 \times 48\pi = 144\pi \). Then height: \( 144\pi = 36\pi \times \text{Height} \implies \text{Height} = 4 \). That matches. So radius is 6, diameter 12, height 4, cylinder volume 144π.

Completed Table:

Diameter (units)	Radius (units)	Base Area (sq. units)	Height (units)	Cylinder Volume (cu. units)	Cone Volume (cu. units)
6	3	\( 9\pi \)	\( \frac{40}{3} \)	\( 120\pi \)	\( 40\pi \)
12	6	\( 36\pi \)	4	\( 144\pi \)	\( 48\pi \)

Key Formulas:

• Radius: \( r = \frac{d}{2} \)
• Base Area (circle): \( A = \pi r^2 \)
• Cylinder Volume: \( V_{\text{cylinder}} = A \times h \)
• Cone Volume: \( V_{\text{cone}} = \frac{1}{3} A \times h \) (or \( V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} \))

Final Answers (for each row):

1. First Row:

• Diameter: \( \boldsymbol{10} \)
• Base Area: \( \boldsymbol{25\pi} \)
• Cylinder Volume: \( \boldsymbol{175\pi} \)
• Cone Volume: \( \boldsymbol{\frac{175\pi}{3}} \)

1. Second Row:

• Radius: \( \boldsymbol{3} \)
• Base Area: \( \boldsymbol{9\pi} \)
• Height: \( \boldsymbol{\frac{40}{3}} \)
• Cylinder Volume: \( \boldsymbol{120\pi} \)

1. Third Row:

• Diameter: \( \boldsymbol{12} \)
• Radius: \( \boldsymbol{6} \)
• Height: \( \boldsymbol{4} \)
• Cylinder Volume: \( \boldsymbol{144\pi} \)

If you need specific values (e.g., numerical approximations), substitute \( \pi \approx 3.14 \) into the expressions.