QQuestionMathematics
QuestionMathematics
find surface area of a triangular prism with measurements of 26 yd, 24 yd, 10 yd, 18 yd.
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Answer
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Step 1:
\begin{align*}
First, we need to find the lengths of the slant height (l) and the height (h) of the triangular faces. We can use the Pythagorean theorem to find l using the given measurements of 26 yd, 24 yd, and 10 yd: l &= \sqrt{26^2 + 24^2} \ &= \sqrt{676 + 576} \ &= \sqrt{1252} \ &\approx 35.38~\text{yd} \end{align*}
Step 2:
\begin{align*}
Now, we can find the height (h) of the triangular face using the given measurement of 10 yd and one of the other two measurements. Let's use 26 yd: h &= \sqrt{26^2 - 10^2} \ &= \sqrt{676 - 100} \ &= \sqrt{576} \ &= 24~\text{yd} \end{align*}
Step 3:
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 26~\text{yd} \times 24~\text{yd} = 312~\text{yd}^2
Next, we need to calculate the area of one of the triangular faces (A) using the formula:
Step 4:
L = 2 \times A = 2 \times 312~\text{yd}^2 = 624~\text{yd}^2
Since a triangular prism has two identical triangular faces, we multiply the area of one face by 2 to find the total lateral surface area (L):
Step 5:
S = L + \text{rectangular face area} = 624~\text{yd}^2 + (18~\text{yd} \times 10~\text{yd}) = 624~\text{yd}^2 + 180~\text{yd}^2
To find the total surface area (S), we also need to calculate the area of the rectangular face, which has a length of 18 yd (the same as the height of the triangular faces) and a width of 10 yd (the base of the triangular faces):
Step 6:
\boxed{S = 804~\text{yd}^2}
Combining the lateral surface area and the rectangular face area, we find the total surface area of the triangular prism:
Final Answer
\boxed{S = 804~\text{yd}^2}
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