Dr. Evers is experimenting with light beams and prisms. He passes a beam of white light through a triangular prism which spreads the light out into its six rainbow colors. The bases of the prism are equilateral triangles. The surface area of this prism is 4,292 square millimeters. The area of each triangular face is 271 square millimeters. Which expression can be used to find h, the height, in millimeters, of the prism?

Rationale

(4,292-2(271))/3(25)

To find the height \( h \) of the prism, we consider the total surface area and the area of the triangular faces. Since the prism has two triangular bases, we subtract the area of these bases from the total surface area before dividing by the perimeter of the triangular base to determine the height.

A (4,292/3(25)) YOUR ANSWER

This expression incorrectly assumes that the total surface area can be directly divided by \( 3(25) \) without accounting for the area of the triangular bases. It does not consider the need to subtract the area of the two triangular faces from the total surface area, leading to an incorrect calculation of height.

B (4,292/271) YOUR ANSWER

While this expression divides the total surface area by the area of one triangular face, it fails to factor in the two triangular bases present in the prism. The height cannot be determined simply by this division, as it overlooks the contribution of both triangular faces to the total surface area.

C ((4,292-271)/25) YOUR ANSWER

This choice subtracts only the area of one triangular face from the total surface area, which is incorrect because there are two triangular bases in the prism. Therefore, this expression does not accurately reflect the geometry of the prism, leading to a miscalculation of the height.

D ((4,292-2(271))/3(25)) CORRECT

This expression correctly accounts for the two triangular bases by subtracting \( 2 \times 271 \) from the total surface area of \( 4,292 \). It then divides by the perimeter of the triangular base, which is \( 3 \times 25 \), accurately yielding the height of the prism.

Conclusion

To find the height \( h \) of the prism, one must consider the total surface area while subtracting the areas of both triangular bases. The correct expression, \( (4,292-2(271))/3(25) \), effectively captures this relationship, allowing for the accurate calculation of the prism's height. Each incorrect choice fails to adequately account for the geometry involved, leading to miscalculations.

(4,292-2(271))/3(25)

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