Question

question
a right triangle is shown. find the value of x. round your answer to the nearest hundredth (2 decimal places).
987.3
15
50
x
answer attempt 45 out of 9999
additional solution no solution
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Explanation:

Step1: Define total base length

Let the base of the larger triangle be $L = 987.3$, the base of the smaller triangle with angle $15^\circ$ be $a$, and the base of the triangle with angle $50^\circ$ be $b$. So $L = a + b$.

Step2: Relate heights to bases

Let the vertical height (common to both triangles) be $h$. For the larger triangle: $h = a \tan(15^\circ)$. For the triangle with $x$: $h = b \tan(50^\circ)$.

Step3: Substitute $a = L - b$

$ (987.3 - b) \tan(15^\circ) = b \tan(50^\circ) $

Step4: Solve for $b$

$987.3 \tan(15^\circ) = b (\tan(50^\circ) + \tan(15^\circ))$
$b = \frac{987.3 \tan(15^\circ)}{\tan(50^\circ) + \tan(15^\circ)}$
Calculate $\tan(15^\circ) \approx 0.2679$, $\tan(50^\circ) \approx 1.1918$
$b = \frac{987.3 \times 0.2679}{1.1918 + 0.2679} = \frac{264.5977}{1.4597} \approx 181.26$

Step5: Solve for $x$

Use $\cos(50^\circ) = \frac{b}{x}$, so $x = \frac{b}{\cos(50^\circ)}$
$\cos(50^\circ) \approx 0.6428$
$x = \frac{181.26}{0.6428} \approx 281.98$

Answer:

281.98