Question

(a) if the volume of the prism on the left is 216 m³, find the volume of the prism on the right.

(b) if the height of the prism on the left is 12 m, find the height of the prism on the right.

(c) if the surface area of the prism on the left is 432 m², find the surface area of the prism on the right.

Explanation:

Part (a)

Step1: Find base area of left prism

The left prism is a rectangular prism with side \( 6 \, \text{m} \) (assuming it's a square base, so length and width are both \( 6 \, \text{m} \)) and volume \( V_1 = 216 \, \text{m}^3 \). The volume of a rectangular prism is \( V = \text{base area} \times \text{height} \). For a square base, base area \( A_1 = 6 \times 6 = 36 \, \text{m}^2 \). Then height \( h_1=\frac{V_1}{A_1}=\frac{216}{36} = 6 \, \text{m} \). Wait, actually, if it's a rectangular prism with length \( l = 6 \), width \( w = 6 \) (since the base is a square as the side is labeled 6m), then base area \( A = l\times w=6\times6 = 36 \, \text{m}^2 \). The right prism has length \( l_2 = 7 \), width \( w_2 = 6 \) (assuming width is same, since only the front side length changes from 6 to 7, so width and height are same for both prisms? Wait, maybe both prisms have the same height and same width, only the length (the front - facing side) changes from 6 to 7. So the base of the left prism: if the front side is 6m (length), and let the width be \( w \) and height be \( h \). Then volume of left prism \( V_1=6\times w\times h = 216 \). For the right prism, volume \( V_2 = 7\times w\times h \). So the ratio of volumes \( \frac{V_2}{V_1}=\frac{7\times w\times h}{6\times w\times h}=\frac{7}{6} \). So \( V_2=V_1\times\frac{7}{6}=216\times\frac{7}{6}=252 \, \text{m}^3 \).

Step2: Calculate volume of right prism

Using the ratio of the lengths (since width and height are constant), the volume of the right prism is \( 216\times\frac{7}{6}=252 \, \text{m}^3 \).

Step1: Analyze height relationship

If the height of the left prism is \( h_1 = 12 \, \text{m} \), and since the prisms have the same width and the only change is in the length (from 6 to 7) but height is a vertical dimension. Wait, no, maybe the height is the same? Wait, no, maybe the prisms are similar in terms of width and height, only the length (the horizontal side) changes. Wait, the problem says "height of the prism", so if the left prism has height \( h_1 = 12 \, \text{m} \), and the right prism has the same height? No, that can't be. Wait, maybe the prisms have the same cross - sectional area (width and height) and only the length (the front - back or left - right side) changes. Wait, the left prism has a side of 6m, right has 7m. If the height is a vertical measurement, and the prisms are such that the height is determined by the same factor? No, actually, if the base of the left prism is a rectangle with length 6 and width \( w \), and height \( h = 12 \). The right prism has length 7, width \( w \), and height \( h_2 \). But if the prisms are "similar" in the sense that the ratio of length is \( \frac{7}{6} \), but height is a vertical dimension. Wait, no, maybe the height is the same? Wait, the problem is probably that the two prisms have the same width and the same height, only the length (the side labeled 6 and 7) changes. So the height of the right prism is the same as the left? No, that doesn't make sense. Wait, maybe I misinterpret. Wait, the left prism: side 6m (let's say length \( l_1 = 6 \)), height \( h_1 = 12 \). The right prism: length \( l_2 = 7 \), and since the prisms are "stretched" along the length, but height is proportional? No, wait, maybe the height is calculated based on the ratio of lengths. Wait, no, the height of a prism is the perpendicular distance between the two bases. If the bases are rectangles with length 6 and 7, and the same width and height (the vertical side) is same? No, that can't be. Wait, maybe the problem is that the two prisms have the same cross - sectional area (width × height) and the length is 6 and 7. So the height of the left prism is 12, and the right prism has the same height? No, that would mean height is 12. But that seems odd. Wait, maybe the prisms are such that the ratio of length is \( \frac{7}{6} \), but height is same. Wait, the question is "find the height of the prism on the right". If the left prism has height 12m, and the prisms have the same width and the length changes from 6 to 7, but height is a vertical dimension, so maybe the height is the same? No, that can't be. Wait, maybe I made a mistake. Wait, the volume of the left prism: if length \( l = 6 \), width \( w \), height \( h = 12 \), then volume \( V = 6\times w\times12 \). But we know from part (a) that volume is 216, so \( 6\times w\times12=216\Rightarrow w\times12 = 36\Rightarrow w = 3 \). Then the right prism: length \( l = 7 \), width \( w = 3 \), height \( h_2 \). But in part (a), volume of right was 252, so \( 7\times3\times h_2=252\Rightarrow h_2=\frac{252}{21}=12 \). Oh! So the height is the same. So the height of the right prism is also 12m? Wait, no, that's the same as left. But that seems correct because when we calculated the volume, the height was 6? Wait, no, in part (a), I think I made a mistake. Let's recalculate part (a) correctly.

Correct part (a): Volume of left prism \( V_1=l_1\times w\times h \), where \( l_1 = 6 \), \( V_1 = 216 \). So \( 6\times w\times h=216\Rightarrow w\times h = 36 \). Volume of right prism \( V_2=l_2\times w\times h=7\times(w\times h)=7\times36 = 252 \),…

Step1: Find surface area formula for rectangular prism

The surface area of a rectangular prism is \( S = 2(lw+lh + wh) \). For the left prism, \( l_1 = 6 \), let \( w = w \), \( h = h \). \( S_1=2(6w + 6h+wh)=432 \). From part (a), we know that \( 6wh = 216\Rightarrow wh = 36 \). Substitute \( wh = 36 \) into \( S_1 \): \( 2(6w + 6h + 36)=432\Rightarrow6w + 6h+36 = 216\Rightarrow6w + 6h=180\Rightarrow w + h = 30 \). For the right prism, \( l_2 = 7 \), \( w \) and \( h \) are same as left (since \( wh = 36 \) and \( w + h = 30 \)). Surface area of right prism \( S_2=2(7w+7h + wh)=2(7(w + h)+wh) \). Substitute \( w + h = 30 \) and \( wh = 36 \): \( S_2=2(7\times30 + 36)=2(210 + 36)=2\times246 = 492 \, \text{m}^2 \).

Step2: Calculate surface area of right prism

Using the formula \( S = 2(lw+lh + wh) \), with \( l = 7 \), \( wh = 36 \), \( w + h = 30 \), we get \( S_2=2(7\times30+36)=492 \, \text{m}^2 \).

Answer:

(a):
\( 252 \)

Part (b)