Question

the oblique pyramid has a square base with an edge length of 5 cm. the height of the pyramid is 7 cm.

what is the volume of the pyramid?

\\(11\frac{2}{3}\text{ cm}^3\\)

\\(43\frac{3}{4}\text{ cm}^3\\)

\\(58\frac{1}{3}\text{ cm}^3\\)

\\(87\frac{1}{2}\text{ cm}^3\\)

Explanation:

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<reasoning>

Calculate the base area of the pyramid

Using the Area of Polygons and Pyramid Base Geometry knowledge points
\[

$$\begin{aligned} s &= 5\text{ cm}\\ B &= s^2 = 5^2 = 25\text{ cm}^2 \end{aligned}$$

\]

Calculate the volume of the oblique pyramid

Using the Volume of Oblique Pyramids knowledge point
The volume \(V\) of any pyramid, whether right or oblique, is given by:
\[
V = \frac{1}{3} \cdot B \cdot h
\]
Substituting the base area \(B = 25\text{ cm}^2\) and height \(h = 7\text{ cm}\):
\[
V = \frac{1}{3} \cdot 25 \cdot 7 = \frac{175}{3}\text{ cm}^3
\]

Convert the volume to a mixed number

Using the Volume of Oblique Pyramids knowledge point
Divide 175 by 3 to find the mixed fraction representation:
\[
175 \div 3 = 58\text{ with a remainder of } 1
\]
\[
V = 58\frac{1}{3}\text{ cm}^3
\]
</reasoning>

<answer>
<mcq-option>(A) \(11\frac{2}{3}\text{ cm}^3\)</mcq-option>
<mcq-option>(B) \(43\frac{3}{4}\text{ cm}^3\)</mcq-option>
<mcq-correct>(C) \(58\frac{1}{3}\text{ cm}^3\)</mcq-correct>
<mcq-option>(D) \(87\frac{1}{2}\text{ cm}^3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Calculate the base area of the pyramid

Using the Area of Polygons and Pyramid Base Geometry knowledge points
\[

$$\begin{aligned} s &= 5\text{ cm}\\ B &= s^2 = 5^2 = 25\text{ cm}^2 \end{aligned}$$

\]

Calculate the volume of the oblique pyramid

Using the Volume of Oblique Pyramids knowledge point
The volume \(V\) of any pyramid, whether right or oblique, is given by:
\[
V = \frac{1}{3} \cdot B \cdot h
\]
Substituting the base area \(B = 25\text{ cm}^2\) and height \(h = 7\text{ cm}\):
\[
V = \frac{1}{3} \cdot 25 \cdot 7 = \frac{175}{3}\text{ cm}^3
\]

Convert the volume to a mixed number

Using the Volume of Oblique Pyramids knowledge point
Divide 175 by 3 to find the mixed fraction representation:
\[
175 \div 3 = 58\text{ with a remainder of } 1
\]
\[
V = 58\frac{1}{3}\text{ cm}^3
\]
</reasoning>

<answer>
<mcq-option>(A) \(11\frac{2}{3}\text{ cm}^3\)</mcq-option>
<mcq-option>(B) \(43\frac{3}{4}\text{ cm}^3\)</mcq-option>
<mcq-correct>(C) \(58\frac{1}{3}\text{ cm}^3\)</mcq-correct>
<mcq-option>(D) \(87\frac{1}{2}\text{ cm}^3\)</mcq-option>
</answer>

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