Question
If the right triangular prism shown is dilated using a scale factor of 3, what would be the volume of the dilated prism? (1 point) Responses 126 cm3 126 cm3 252 cm3 252 cm3 1,134 cm3 1,134 cm3 2,268 cm3. L 7cm W 4cm H 3cm. Choose one answer even if wrong
Answers
GPT-4o mini
To determine the volume of a prism, we use the formula:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
For a right triangular prism, the base area can be calculated using the area of the triangle that forms its base. The area \(A\) of a right triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the dimensions of the triangular base are given as length \(L = 7 \, \text{cm}\) and width \(W = 4 \, \text{cm}\). Thus, the area of the base triangle is:
\[
A = \frac{1}{2} \times 7 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 28 \, \text{cm}^2 = 14 \, \text{cm}^2
\]
Now, the height of the prism is given as \(H = 3 \, \text{cm}\). Therefore, the volume of the prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height} = 14 \, \text{cm}^2 \times 3 \, \text{cm} = 42 \, \text{cm}^3
\]
Now, when the prism is dilated using a scale factor of 3, the volume of the prism will increase by the cube of the scale factor:
\[
\text{New Volume} = \text{Original Volume} \times (\text{Scale Factor})^3 = 42 \, \text{cm}^3 \times 3^3 = 42 \, \text{cm}^3 \times 27 = 1134 \, \text{cm}^3
\]
Thus, the volume of the dilated prism is \(1,134 \, \text{cm}^3\).
So, the answer is **1,134 cm³**.
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
For a right triangular prism, the base area can be calculated using the area of the triangle that forms its base. The area \(A\) of a right triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the dimensions of the triangular base are given as length \(L = 7 \, \text{cm}\) and width \(W = 4 \, \text{cm}\). Thus, the area of the base triangle is:
\[
A = \frac{1}{2} \times 7 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 28 \, \text{cm}^2 = 14 \, \text{cm}^2
\]
Now, the height of the prism is given as \(H = 3 \, \text{cm}\). Therefore, the volume of the prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height} = 14 \, \text{cm}^2 \times 3 \, \text{cm} = 42 \, \text{cm}^3
\]
Now, when the prism is dilated using a scale factor of 3, the volume of the prism will increase by the cube of the scale factor:
\[
\text{New Volume} = \text{Original Volume} \times (\text{Scale Factor})^3 = 42 \, \text{cm}^3 \times 3^3 = 42 \, \text{cm}^3 \times 27 = 1134 \, \text{cm}^3
\]
Thus, the volume of the dilated prism is \(1,134 \, \text{cm}^3\).
So, the answer is **1,134 cm³**.