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Height From a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are
35^(@) and
47^(@)40^('), respectively.
(a) Draw right triangles that give a visual representation of the problem. Label the known quantities and the unknown height of the steeple.
(b) Use a trigonometric function to write an equation involving the unknown.
(c) Find the height of the steeple.

Height From a point $50$ feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are $35^{\circ}$ and $47^{\circ} 40^{\prime}$ , respectively. $\newline$ (a) Draw right triangles that give a visual representation of the problem. Label the known quantities and the unknown height of the steeple. $\newline$ (b) Use a trigonometric function to write an equation involving the unknown. $\newline$ (c) Find the height of the steeple.

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Area of quadrilaterals and triangles: word problems

Full solution

Q. Height From a point

50

feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are

35^{\circ}

and

47^{\circ} 40^{\prime}

, respectively.

\newline

(a) Draw right triangles that give a visual representation of the problem. Label the known quantities and the unknown height of the steeple.

\newline

(b) Use a trigonometric function to write an equation involving the unknown.

\newline

(c) Find the height of the steeple.

Draw Triangles: (a) To draw the triangles, sketch two right triangles sharing a common base, which is the distance from the observation point to the church ( $50$ feet). The first triangle includes the angle of elevation to the base of the steeple ( $35$ degrees), and the second, larger triangle includes the angle of elevation to the top of the steeple ( $47$ degrees $40$ minutes). The height of the steeple is the difference in the heights of the two triangles.
Use Tangent Function: (b) Use the tangent function, which relates the angle of elevation to the opposite side (height) and adjacent side (distance from the church). For the smaller triangle, $\tan(35^\circ) = \frac{\text{height of base}}{50 \text{ feet}}$ . For the larger triangle, $\tan(47^\circ 40') = \frac{\text{total height}}{50 \text{ feet}}$ .
Convert to Decimal Form: (c) First, convert $47$ degrees $40$ minutes to decimal form. There are $60$ minutes in a degree, so $40$ minutes is $\frac{40}{60}$ degrees, which is $\frac{2}{3}$ degrees. Therefore, $47$ degrees $40$ minutes is $47 + \frac{2}{3}$ degrees, or $47.6667$ degrees. Now, calculate the height of the base using the tangent of $40$ $0$ degrees: $40$ $1$ . Height of base = $40$ $2$ . Calculate the total height using the tangent of $47.6667$ degrees: $40$ $4$ . Total height = $40$ $5$ .

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