The bases of the prism are equilateral triangles. The surface area of this prism is 4,292. Which expression can be used to find [tex]$h$[/tex], the height, in millimeters, of the prism?
[tex]$\begin{array}{l}
\frac{4,292}{271} \
\frac{4,292-2(271)}{3(25)} \
\frac{4,292-271}{25} \
\frac{4,292}{3(25)}
\end{array}$[/tex]
Answer (2)
The problem involves finding the correct expression for the height of a prism with equilateral triangle bases, given its surface area.
We start by recalling the formula for the surface area of such a prism: A = 2 × Area of base + 3 × ( s × h ) .
We isolate h in the formula to get: h = 3 s 4292 − 2 s 2 3 .
By assuming the area of the base is 271 and the side length is 25, we find the correct expression: 3 ( 25 ) 4292 − 2 ( 271 ) .
Explanation
Problem Analysis Let's analyze the problem. We have a prism with equilateral triangle bases and a given surface area of 4292 square millimeters. We need to find the expression that correctly calculates the height, h , of the prism.
Surface Area Formula The surface area of a prism with equilateral triangle bases is given by the formula: A = 2 × Area of base + 3 × ( s × h ) , where s is the side length of the equilateral triangle and h is the height of the prism. The area of an equilateral triangle with side s is 4 s 2 3 . So, the surface area can be written as: A = 2 × 4 s 2 3 + 3 s h = 2 s 2 3 + 3 s h .
Isolating h We are given that the total surface area A = 4292 . So, we have: 4292 = 2 s 2 3 + 3 s h .
We need to isolate h to find the correct expression. First, subtract 2 s 2 3 from both sides: 4292 − 2 s 2 3 = 3 s h .
Now, divide by 3 s to solve for h :
h = 3 s 4292 − 2 s 2 3 = 6 s 8584 − s 2 3 .
Evaluating Options Now, let's examine the given options. The options suggest that the area of the equilateral triangle is approximately 271, and the side length s is approximately 25. If we assume that the area of each equilateral triangle base is 271, then the combined area of the two bases is 2 × 271 = 542 .
The lateral surface area is then 4292 − 542 = 3750 .
If we assume that each side of the equilateral triangle is approximately 25, then the lateral surface area is 3 × 25 × h = 75 h .
Thus, 75 h = 3750 , so h = 75 3750 = 50 .
Now, let's check the given options to see which one gives us h = 50 :
271 4292 ≈ 15.84 , which is not 50. 3 ( 25 ) 4292 − 2 ( 271 ) = 75 4292 − 542 = 75 3750 = 50 . This matches our calculation. 25 4292 − 271 = 25 4021 = 160.84 , which is not 50. 3 ( 25 ) 4292 = 75 4292 ≈ 57.23 , which is not 50.
Final Answer Therefore, the correct expression to find h is 3 ( 25 ) 4292 − 2 ( 271 ) .
Examples
Imagine you are designing a tent in the shape of a triangular prism. You know the total surface area of the tent material you have and the approximate area of the triangular base. Using the formula we derived, you can calculate the height of the tent needed to achieve the desired surface area. This ensures you have enough space inside the tent while utilizing the available material efficiently.
To find the height h of the prism, use the expression 3 ( 25 ) 4292 − 2 ( 271 ) . This corresponds to the surface area of the prism given that the area of each triangular base is assumed to be 271 and the side length is 25. Therefore, the correct choice is the second option.
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