Question

consider the following right triangle and the work shown to solve for x.
image of right triangle with one leg 15, hypotenuse 25, angle 53.13° at the top, and the other leg x
\\(\tan(53.13^{\circ}) = \frac{x}{25}\\)
\\(25\tan(53.13^{\circ}) = x\\)
\\(33.3 \approx x\\)
what statement best describes the the work shown?
\\(\bigcirc\\) the proportion was set up incorrectly
\\(\bigcirc\\) the proportion was solved incorrectly.
\\(\bigcirc\\) there was no error made.
correctly find the value of x. round your answer to one decimal place. if there was no error, enter the value for x from the work above.

Explanation:

Step1: Analyze the right triangle

In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. For the given angle \(53.13^\circ\), the opposite side is \(x\) and the adjacent side is \(15\), not \(25\). So the correct tangent ratio should be \(\tan(53.13^\circ)=\frac{x}{15}\), not \(\frac{x}{25}\). Thus, the proportion was set up incorrectly.

Step2: Solve for \(x\) correctly

Using the correct ratio \(\tan(53.13^\circ)=\frac{x}{15}\), we can solve for \(x\) by multiplying both sides by \(15\). So \(x = 15\times\tan(53.13^\circ)\). We know that \(\tan(53.13^\circ)\approx\frac{4}{3}\) (since \(\tan(53^\circ)\approx\frac{4}{3}\) approximately). So \(x = 15\times\frac{4}{3}=20.0\). Also, using a calculator, \(\tan(53.13^\circ)\approx1.333\), then \(x = 15\times1.333\approx20.0\).

Answer:

The proportion was set up incorrectly. The correct value of \(x\) is \(20.0\).