Question

right rectangular prism x is similar to right rectangular prism y. the surface area of right rectangular prism x is 56 square centimeters (cm²), and the surface area of right rectangular prism y is 1,400 cm². the volume of right rectangular prism y is 3,000 cubic centimeters (cm³). what is the sum of the volumes, in cm³, of right rectangular prism x and right rectangular prism y?
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Explanation:

Step1: Find the scale factor of surface areas

For similar solids, the ratio of their surface areas is the square of the scale factor (\(k\)). Let the surface area of prism X be \(SA_X = 56\) and prism Y be \(SA_Y=1400\). Then \(\frac{SA_X}{SA_Y}=k^{2}\), so \(k^{2}=\frac{56}{1400}=\frac{1}{25}\), and \(k = \frac{1}{5}\) (since scale factor is positive).

Step2: Find the volume of prism X

The ratio of volumes of similar solids is the cube of the scale factor. Let volume of X be \(V_X\) and Y be \(V_Y = 3000\). So \(\frac{V_X}{V_Y}=k^{3}\). Substituting \(k=\frac{1}{5}\), we get \(\frac{V_X}{3000}=(\frac{1}{5})^{3}=\frac{1}{125}\). Solving for \(V_X\), we have \(V_X=\frac{3000}{125}=24\).

Step3: Find the sum of volumes

Sum \(= V_X + V_Y=24 + 3000 = 3024\).

Answer:

\(3024\)