Q4. Maria wants to determine the height, $$ x $$, of a nearby tree. She stands 53 feet from the base of the tree. The measure of the angle of elevation from Maria to the top of the tree is $$ 46^{\circ} $$. Select ALL of the following that can be used to find the height of the tree. $$ x=53 \sin 46^{\circ} $$ $$ x=\frac{53}{\sin 46^{\circ}} $$ $$ x=53 \cos 46^{\circ} $$ $$ x=\frac{53}{\cos 46^{\circ}} $$ $$ x=53 an 46^{\circ} $$ $$ x=\frac{53}{ an 46^{\circ}} $$ Show your work.

Question

Q4. Maria wants to determine the height, $$ x $$, of a nearby tree. She stands 53 feet from the base of the tree. The measure of the angle of elevation from Maria to the top of the tree is $$ 46^{\circ} $$. Select ALL of the following that can be used to find the height of the tree. $$ x=53 \sin 46^{\circ} $$ $$ x=\frac{53}{\sin 46^{\circ}} $$ $$ x=53 \cos 46^{\circ} $$ $$ x=\frac{53}{\cos 46^{\circ}} $$ $$ x=53 	an 46^{\circ} $$ $$ x=\frac{53}{	an 46^{\circ}} $$ Show your work.

UpStudy AI · Accepted Answer

We first draw the right triangle from the situation. Maria stands 53 feet away from the tree, so the horizontal leg of the triangle is 53 feet. The tree’s height is the vertical leg, $$ x $$. The angle of elevation from Maria to the top of the tree is $$ 46^\circ $$.

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Thus, we have

$$
\tan 46^\circ=\frac{x}{53}.
$$

To solve for $$ x $$, multiply both sides of the equation by 53:

$$
x=53\tan46^\circ.
$$

Now, we review the options provided:

1. $$ x=53\sin46^\circ $$  
   This is incorrect because the sine of $$ 46^\circ $$ represents the ratio of the opposite side to the hypotenuse, not the adjacent side.

2. $$ x=\frac{53}{\sin46^\circ} $$  
   This is incorrect because it would give the length of the hypotenuse, not the height $$ x $$.

3. $$ x=53\cos46^\circ $$  
   This is incorrect because the cosine of $$ 46^\circ $$ represents the ratio of the adjacent side to the hypotenuse.

4. $$ x=\frac{53}{\cos46^\circ} $$  
   This is incorrect because it calculates the hypotenuse, not $$ x $$.

5. $$ x=53\tan46^\circ $$  
   This is correct as we derived above.

6. $$ x=\frac{53}{\tan46^\circ} $$  
   This is incorrect since the correct equation is $$ x=53\tan46^\circ $$, not the reciprocal.

Thus, the only choice that can be used to find the height of the tree is

$$
x=53\tan46^\circ.
$$
The correct formula to find the height of the tree is $$ x = 53 \tan 46^\circ $$.

Q4.
Maria wants to determine the height, , of a nearby tree. She stands 53 feet from the
base of the tree. The measure of the angle of elevation from Maria to the top of the
tree is .
Select ALL of the following that can be used to find the height of the tree.

Show your work.

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Q4. Maria wants to determine the height, , of a nearby tree. She stands 53 feet from the base of the tree. The measure of the angle of elevation from Maria to the top of the tree is . Select ALL of the following that can be used to find the height of the tree. Show your work.