Question

two similar rectangular prisms have surface areas of 112 square centimeters and 1008 square centimeters. if the length and width of the base of the smaller prism are 4 centimeters and 2 centimeters respectively, what is the perimeter of one base of the larger prism?
_____ cm

Explanation:

Step1: Find the height of the smaller prism

First, we need to find the height of the smaller rectangular prism. The surface area of a rectangular prism is given by the formula \( SA = 2(lw + lh + wh) \), where \( l \) is length, \( w \) is width, and \( h \) is height. For the smaller prism, \( l = 4 \) cm, \( w = 2 \) cm, and \( SA = 112 \) cm². Plugging these values into the formula:

\( 112 = 2((4)(2) + (4)h + (2)h) \)

Simplify inside the parentheses:

\( 112 = 2(8 + 4h + 2h) \)

\( 112 = 2(8 + 6h) \)

Divide both sides by 2:

\( 56 = 8 + 6h \)

Subtract 8 from both sides:

\( 48 = 6h \)

Divide by 6:

\( h = 8 \) cm

Step2: Find the scale factor of the surface areas

The surface areas of the two similar prisms are 112 cm² (smaller) and 1008 cm² (larger). The ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding linear dimensions (scale factor \( k \)). So:

\( \frac{SA_{larger}}{SA_{smaller}} = k^2 \)

\( \frac{1008}{112} = k^2 \)

\( 9 = k^2 \)

Taking the square root of both sides (since scale factor is positive):

\( k = 3 \)

Step3: Find the length and width of the larger prism's base

The scale factor \( k = 3 \) means the linear dimensions of the larger prism are 3 times those of the smaller. The smaller prism's base has \( l = 4 \) cm and \( w = 2 \) cm. So for the larger prism:

\( l_{larger} = 4 \times 3 = 12 \) cm

\( w_{larger} = 2 \times 3 = 6 \) cm

Step4: Calculate the perimeter of the larger base

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). For the larger base:

\( P = 2(12 + 6) \)

\( P = 2(18) \)

\( P = 36 \) cm

Answer:

36