Question

applying the 30°-60°-90° triangle theorem
a salt barn is used by a highway maintenance company to store road salt for use during freezing weather. the barn is in the shape of a triangular prism with an equilateral triangle as its front face. the distance from the peak of the barn to the ground, along the roofline, is 30 feet. how tall is the salt barn?
image of a triangle with height h, angle 60°, side 30
\\(\frac{2}{\sqrt{3}} = \frac{}{h}\\)
\\(h = 15\sqrt{3} \approx\\) ____

• check to verify that the ____ of the barn

is smaller than the ____ by using a
calculator.

• 26 is indeed smaller than ____. because the

hypotenuse really is the ____ side, this
does verify our result.

Explanation:

Step1: Calculate \( 15\sqrt{3} \)

We know that \( \sqrt{3} \approx 1.732 \). So, \( 15\sqrt{3}=15\times1.732 = 25.98\approx26 \) (rounded to a reasonable decimal place).

Step2: Analyze the triangle properties

In a \( 30^{\circ}-60^{\circ}-90^{\circ} \) triangle (which is half of the equilateral triangle here), the hypotenuse is the longest side. The height \( h \) of the barn (the altitude of the equilateral triangle, which is the side opposite the \( 60^{\circ} \) angle in the \( 30 - 60 - 90 \) triangle) should be smaller than the hypotenuse (the roofline, which is 30 feet). We check that \( 26 \) (approximate height) is smaller than \( 30 \) (the hypotenuse length), and since the hypotenuse is the longest side in a right - triangle, this makes sense.

Answer:

The height of the salt barn is approximately \( 15\sqrt{3}\approx26 \) feet. For the blanks: height, roofline (hypotenuse), 30, longest.