Question

a student is standing 20 feet away from the base of a tree. he looks to the top of the tree at a 50° angle of elevation. his eyes are 5 feet above the ground. using cos 50° ≈ 0.64, what is the height of the tree to the nearest foot? 31 foot 24 foot 36 foot 29 foot

Explanation:

Step1: Find the adjacent and opposite sides relation

We know that in a right triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the adjacent side (distance from student to tree base horizontally) is 20 feet, the angle of elevation $\theta = 50^{\circ}$, and we need to find the opposite side (height from eyes to tree top), let's call it $h$. First, we can find $\sin\theta$ using the identity $\sin^{2}\theta+\cos^{2}\theta = 1$, but wait, actually we can use $\tan\theta=\frac{\sin\theta}{\cos\theta}$, but maybe easier to first find $\sin\theta$. Wait, alternatively, we can use $\tan\theta=\frac{h}{20}$, but we know $\cos50^{\circ}\approx0.64$, so $\sin50^{\circ}=\sqrt{1 - \cos^{2}50^{\circ}}\approx\sqrt{1 - 0.64^{2}}=\sqrt{1 - 0.4096}=\sqrt{0.5904}\approx0.768$. Then $\tan50^{\circ}=\frac{\sin50^{\circ}}{\cos50^{\circ}}\approx\frac{0.768}{0.64} = 1.2$. Then $h = 20\times\tan50^{\circ}\approx20\times1.2 = 24$? Wait, no, wait, maybe I made a mistake. Wait, actually, the height from eyes to tree top: let's denote the height from ground to eyes as 5 ft, and from eyes to tree top as $y$. Then the total height $H=5 + y$. We have a right triangle with adjacent side 20 ft, angle $50^{\circ}$, so $\tan(50^{\circ})=\frac{y}{20}$, so $y = 20\times\tan(50^{\circ})$. We know that $\tan(50^{\circ})\approx1.1918$, so $y\approx20\times1.1918 = 23.836$. Then total height $H=5 + 23.836\approx28.836\approx29$? Wait, but wait, maybe the problem expects using $\cos$ to find the hypotenuse first? Wait, the problem says "using $\cos50^{\circ}\approx0.64$". So let's do that. The adjacent side is 20 ft, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, so hypotenuse $c=\frac{20}{\cos50^{\circ}}\approx\frac{20}{0.64}=31.25$ ft. Then, using $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, so opposite side (height from eyes to tree top) $y = c\times\sin50^{\circ}$. We can find $\sin50^{\circ}$ as $\sqrt{1 - 0.64^{2}}\approx0.768$, so $y\approx31.25\times0.768 = 24$? Wait, no, 31.250.768: 310.768=23.808, 0.250.768=0.192, total 24. So then total height is 5 + 24 = 29? Wait, no, 20/cos50 is 31.25, then sin50 is approx 0.768, so 31.250.768=24, then 24 +5=29. Yes, that makes sense. Let's recheck:

Step1: Find hypotenuse using cosine

$\cos(50^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{20}{c}$, so $c=\frac{20}{\cos(50^{\circ})}\approx\frac{20}{0.64}=31.25$ ft.

Step2: Find opposite side (height from eyes to tree top) using sine

$\sin(50^{\circ})=\frac{\text{opposite}}{c}=\frac{y}{31.25}$, so $y = 31.25\times\sin(50^{\circ})$. Since $\sin^{2}\theta+\cos^{2}\theta = 1$, $\sin(50^{\circ})=\sqrt{1 - 0.64^{2}}=\sqrt{1 - 0.4096}=\sqrt{0.5904}\approx0.768$. Then $y\approx31.25\times0.768 = 24$ ft.

Step3: Find total height of tree

Total height $H = 5 + y = 5 + 24 = 29$ ft.

Answer:

29 feet