Question

the pair of solids is similar. find the volume of each solid. round to the nearest tenth, if necessary.
volume of small solid =
cm³
volume of large solid =
cm³

Explanation:

Step1: Identify the shape (cylinder)

The solids are cylinders. The formula for the volume of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Step2: Volume of the small (blue) cylinder

For the blue cylinder: radius \( r = 5 \, \text{cm} \) (wait, no, looking at the diagram, the blue cylinder has height \( h = 5 \, \text{cm} \), and the radius? Wait, the green cylinder has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \)? Wait, maybe I misread. Wait, the blue cylinder: let's check the dimensions. Wait, the blue cylinder: height \( h = 5 \, \text{cm} \), and the radius? Wait, the green cylinder (large) has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \), and the blue (small) has radius... Wait, maybe the blue cylinder has radius \( r \) and height \( 5 \, \text{cm} \), and green has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \)? Wait, no, maybe the blue cylinder has radius \( 5 \, \text{cm} \)? Wait, no, the diagram: blue cylinder, height \( 5 \, \text{cm} \), and the green cylinder, height \( 10 \, \text{cm} \), radius \( 5 \, \text{cm} \)? Wait, no, maybe the blue cylinder has radius \( r \) and height \( 5 \, \text{cm} \), and green has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \)? Wait, no, let's re-express. Wait, the volume of a cylinder is \( V = \pi r^2 h \).

Wait, maybe the blue cylinder (small) has radius \( r_1 \) and height \( h_1 = 5 \, \text{cm} \), and the green (large) has radius \( r_2 = 5 \, \text{cm} \) and height \( h_2 = 10 \, \text{cm} \)? No, that can't be. Wait, maybe the blue cylinder has radius \( 5 \, \text{cm} \)? Wait, no, the diagram: the blue cylinder's height is \( 5 \, \text{cm} \), and the green's height is \( 10 \, \text{cm} \), and the green's radius is \( 5 \, \text{cm} \). Wait, maybe the blue cylinder has radius \( r \) and height \( 5 \, \text{cm} \), and green has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \). Wait, no, maybe the blue cylinder has radius \( 5 \, \text{cm} \)? Wait, I think I made a mistake. Let's look again. The blue cylinder: height \( 5 \, \text{cm} \), and the green cylinder: height \( 10 \, \text{cm} \), radius \( 5 \, \text{cm} \). Wait, no, the green cylinder's radius is \( 5 \, \text{cm} \), height \( 10 \, \text{cm} \). The blue cylinder: let's assume the radius is \( r \), but maybe the blue cylinder has radius \( 5 \, \text{cm} \) and height \( 5 \, \text{cm} \)? No, that doesn't make sense. Wait, maybe the blue cylinder is the small one with height \( 5 \, \text{cm} \) and radius \( 5 \, \text{cm} \)? No, the green is larger. Wait, the ratio of heights: blue height \( 5 \, \text{cm} \), green height \( 10 \, \text{cm} \), so scale factor \( k = \frac{10}{5} = 2 \)? Wait, no, if they are similar, the ratio of corresponding linear dimensions is the scale factor. Wait, maybe the blue cylinder has radius \( r \) and height \( 5 \, \text{cm} \), and green has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \). Wait, no, maybe the blue cylinder has radius \( 5 \, \text{cm} \) and height \( 5 \, \text{cm} \), and green has radius \( 5 \, \text{cm} \) and height \( 10 \, \text{cm} \)? No, that would make the green taller, but radius same. Wait, no, the volume of a cylinder is \( V = \pi r^2 h \). Let's check the green cylinder: radius \( 5 \, \text{cm} \), height \( 10 \, \text{cm} \). So volume \( V_{green} = \pi (5)^2 (10) = 250\pi \approx 785.4 \, \text{cm}^3 \). The blue cylinder: height \( 5 \, \text{cm} \), radius? Wait, maybe the blue cylinder has radius \( 5 \, \text{cm} \) and height \( 5 \, \text{cm} \)? No, that would be \( V_{blue} = \pi (5)^2 (5) = 125\pi \approx 392.7 \, \text{cm}^3 \). But wait, the problem says "the pair of solids is similar". So similar cylinders: the ratio of radii is equal to the ratio of heights. Let's denote the small (blue) cylinder: height \( h_1 = 5 \, \text{cm} \), radius \( r_1 \). Large (green) cylinder: height \( h_2 = 10 \, \text{cm} \), radius \( r_2 = 5 \, \text{cm} \). Since they are similar, \( \frac{r_1}{r_2} = \frac{h_1}{h_2} \), so \( \frac{r_1}{5} = \frac{5}{10} \), so \( r_1 = 2.5 \, \text{cm} \). Ah, that makes sense! So small cylinder: radius \( 2.5 \, \text{cm} \), height \( 5 \, \text{cm} \). Large cylinder: radius \( 5 \, \text{cm} \), height \( 10 \, \text{cm} \). Now calculate volumes.

Step3: Volume of small (blue) cylinder

\( V_1 = \pi r_1^2 h_1 = \pi (2.5)^2 (5) \)
First, \( (2.5)^2 = 6.25 \), then \( 6.25 \times 5 = 31.25 \), so \( V_1 = 31.25\pi \approx 31.25 \times 3.1416 \approx 98.175 \approx 98.2 \, \text{cm}^3 \)? Wait, no, wait, \( 31.25\pi \approx 98.17477 \approx 98.2 \, \text{cm}^3 \). Wait, but maybe I messed up the radius. Wait, the green cylinder has radius \( 5 \, \text{cm} \), height \( 10 \, \text{cm} \). The blue cylinder: height \( 5 \, \text{cm} \), so scale factor \( k = \frac{5}{10} = 0.5 \). So radius of blue is \( 5 \times 0.5 = 2.5 \, \text{cm} \), height \( 10 \times 0.5 = 5 \, \text{cm} \). So volume of small: \( \pi (2.5)^2 (5) = \pi \times 6.25 \times 5 = 31.25\pi \approx 98.2 \, \text{cm}^3 \). Volume of large: \( \pi (5)^2 (10) = 250\pi \approx 785.4 \, \text{cm}^3 \). Wait, but let's check again. Wait, the green cylinder's radius is \( 5 \, \text{cm} \), height \( 10 \, \text{cm} \). So volume \( V_{large} = \pi r^2 h = \pi (5)^2 (10) = 250\pi \approx 785.4 \, \text{cm}^3 \). The small cylinder: height \( 5 \, \text{cm} \), radius \( 2.5 \, \text{cm} \) (since similar, ratio 1:2 in height, so radius ratio 1:2). So volume \( V_{small} = \pi (2.5)^2 (5) = \pi \times 6.25 \times 5 = 31.25\pi \approx 98.2 \, \text{cm}^3 \).

Wait, but maybe the blue cylinder is the large one? No, the green is taller. Wait, the diagram: blue cylinder is above, green below. Blue has height \( 5 \, \text{cm} \), green has height \( 10 \, \text{cm} \), and green has radius \( 5 \, \text{cm} \). So blue is small, green is large. So:

Volume of small solid (blue cylinder):

Step1: Identify formula for cylinder volume

The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height.

Step2: Determine dimensions of small cylinder

For the small (blue) cylinder:
- Height \( h_1 = 5 \, \text{cm} \)
- Since the solids are similar, the ratio of radii equals the ratio of heights. The large (green) cylinder has radius \( r_2 = 5 \, \text{cm} \) and height \( h_2 = 10 \, \text{cm} \). Thus, the scale factor for linear dimensions is \( \frac{h_1}{h_2} = \frac{5}{10} = 0.5 \).
- Radius of small cylinder: \( r_1 = r_2 \times 0.5 = 5 \times 0.5 = 2.5 \, \text{cm} \).

Step3: Calculate volume of small cylinder

Substitute \( r_1 = 2.5 \, \text{cm} \) and \( h_1 = 5 \, \text{cm} \) into the volume formula:
\[ V_{\text{small}} = \pi (2.5)^2 (5) \]
First, calculate \( (2.5)^2 = 6.25 \). Then:
\[ V_{\text{small}} = \pi \times 6.25 \times 5 = 31.25\pi \approx 31.25 \times 3.1416 \approx 98.2 \, \text{cm}^3 \]

Step4: Determine dimensions of large cylinder

For the large (green) cylinder:
- Radius \( r_2 = 5 \, \text{cm} \)
- Height \( h_2 = 10 \, \text{cm} \)

Step5: Calculate volume of large cylinder

Substitute \( r_2 = 5 \, \text{cm} \) and \( h_2 = 10 \, \text{cm} \) into the volume formula:
\[ V_{\text{large}} = \pi (5)^2 (10) \]
Calculate \( (5)^2 = 25 \). Then:
\[ V_{\text{large}} = \pi \times 25 \times 10 = 250\pi \approx 250 \times 3.1416 \approx 785.4 \, \text{cm}^3 \]

Answer:

Volume of small solid = \( \boldsymbol{98.2} \, \text{cm}^3 \) (rounded to nearest tenth)
Volume of large solid = \( \boldsymbol{785.4} \, \text{cm}^3 \) (rounded to nearest tenth)