One end of the glass prism is in the shape a triangle with a height that is 1 inch longer than twice the base. If the area of the triangle is 39 square inches, then how long are the base and height?

Answer 1

Greetings! Lets solve this shall we ?

So, we must find the dimensions of the height and base of the triangular prism given the area of a triangle is 39 square inches. Also, the height is 1 inch longer than twice the base. The translation for this is

h = 2x + 1

b = x

Then, the area of a triangle is A = (1/2)bh = (1/2)(x)(2x+1).........We then begin simplifying such that

39 = (1/2)(x)(2x+1)............Multiply each side by 2 to get

78 = 2x² + x.........After distributing the x inside......Then we subtract 78 to the right side such that

2x² + x - 78 = 0.........We must factor this to find the base and height. We notice this can factor into

(2x + 13)(x - 6) = 0.......Applying the Zero-Product Property of Equality, we get

2x + 13 = 0 and x - 6 = 0.........We solve for x for both factors to obtain

x = -13/2 and x = 6.........We cant use -13/2 because we cannot have a negative length so we use x = 6. Finally, the base is x = b = 6 inches.

Then, the height is h = 2(6) + 1 = 13 inches

I hope this helped!