A trophy consists of a right triangle on top of a rectangular pedestal. The height of the triangle is equal to the length of the pedestal, and the length of the base of the triangle is 2 centimeters less than the length of the pedestal. If the area of the triangle and the pedestal are equal and the height of the pedestal is one centimeter, how tall is the combined trophy?

Question

High School

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Answer :

LucasMatthewHarris LucasMatthewHarris · Answer 1 · 2024-10-18T05:56:27-04:00

To solve this problem, we need to find the height of the entire trophy, which consists of a right triangle on top of a rectangular pedestal.

Let's denote the length of the rectangular pedestal as [tex]l[/tex] centimeters.

According to the problem, the height of the pedestal is 1 centimeter.
The height of the triangle is equal to the length of the pedestal, so the height of the triangle is also [tex]l[/tex] centimeters.
The base of the triangle is 2 centimeters less than the length of the pedestal, so the base of the triangle is [tex]l - 2[/tex] centimeters.

The area of a rectangle is given by [tex]\text{Area} = \text{length} \times \text{height}[/tex]. Thus, the area of the rectangular pedestal is:

[tex]A_{\text{pedestal}} = l \times 1 = l[/tex]

The area of a triangle is given by [tex]\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}[/tex]. So, the area of the triangle is:

[tex]A_{\text{triangle}} = \frac{1}{2} \times (l - 2) \times l[/tex]

The problem states that the area of the triangle and the pedestal are equal:

[tex]l = \frac{1}{2} \times (l - 2) \times l[/tex]

Multiply both sides by 2 to eliminate the fraction:

[tex]2l = (l - 2)l[/tex]

Simplify and solve for [tex]l[/tex]:

[tex]2l = l^2 - 2l[/tex]

Add [tex]2l[/tex] to both sides:

[tex]4l = l^2[/tex]

Rearrange the equation:

[tex]l^2 - 4l = 0[/tex]

Factor the equation:

[tex]l(l - 4) = 0[/tex]

This gives two possible solutions: [tex]l = 0[/tex] or [tex]l = 4[/tex]. Since a length cannot be zero, we have [tex]l = 4[/tex] centimeters.

Now, let's determine the total height of the trophy:

The total height of the trophy is:

[tex]\text{Total height} = 1 + 4 = 5 \text{ centimeters}[/tex]

Thus, the combined height of the trophy is 5 centimeters.