A tree on a hillside casts a shadow c ft down the hill. If the angle of inclination of the hillside is b to the horizontal and the angle of elevation of the sun is a, find the height of the tree. Assume a = 48°, b = 18° c = 235 ft. (Round your answer to the nearest whole number.)
draw a diagram. You will see that
(h + c sin b)/(c cos b) = tan a
so,
(h + 235 sin 18°)/(235 cos 18°) = tan 48°
h = 175.6 ft
Why did the tree go to school? To get a little 'trunk' education! Okay, let's tackle this problem. We need to find the height of the tree, so let's call that 'h'. Now, we have the angle of elevation of the sun 'a', which is 48°. And we have the angle of inclination of the hillside 'b', which is 18°. Lastly, we have the length of the shadow 'c', which is 235 ft.
Now, imagine a right triangle formed by the tree, its shadow, and the hillside. The height of the tree 'h' is the opposite side to the angle 'a'. The length of the shadow 'c' is the adjacent side to the angle 'a'. And the hillside acts as the hypotenuse.
Using trigonometry, we can use the tangent function to find the height 'h'. The tangent of an angle is equal to the opposite side divided by the adjacent side. So, in this case, we have:
tan(a) = h/c
Plugging in the values, we get:
tan(48°) = h/235
Now, let's solve for 'h'. Multiply both sides by 235:
235 * tan(48°) = h
Using a calculator, we get:
h ≈ 280 ft
So, the height of the tree is approximately 280 feet.
To find the height of the tree, we can use trigonometry and set up a right triangle.
Let's label the sides of the triangle:
- The vertical side (height of the tree) as h.
- The horizontal side (base of the shadow) as x.
- The hypotenuse (the distance from the top of the tree to the tip of the shadow) as d.
We can use the trigonometric ratios sine and tangent to relate the angles and sides:
From the angle of inclination b to the horizontal:
tangent(b) = h/x (equation 1)
From the angle of elevation a, we can use the complementary angle (90 - a) to relate it to angle b:
b = 90 - a (equation 2)
From equation 2, we can substitute the value of b in equation 1:
tangent(90 - a) = h/x
We can use the property that tangent(90 - a) = cotangent(a):
cotangent(a) = h/x (equation 3)
Now, let's find the values of cotangent(a):
cotangent(48°) = h/x
To solve for x, we can rearrange equation 3:
x = h / cotangent(a)
Now, let's substitute the given values:
a = 48°
b = 18°
c = 235 ft
We want to find the height of the tree, h.
Let's plug in the values and calculate:
cotangent(48°) = h / c
cotangent(48°) ≈ 0.9031
Solving for h:
h = cotangent(a) * c
h = 0.9031 * 235
h ≈ 211.84
Rounding to the nearest whole number, the height of the tree is 212 feet.
To find the height of the tree, we can use trigonometric ratios and the given information.
Let's label the height of the tree as "h" and the distance from the base of the tree to the point where the shadow intersects the hillside as "d."
First, draw a diagram to visualize the situation. It helps to draw a right triangle representing the hillside with the base being the distance d, the height being c, and the angle of inclination b.
Now, since we know the angle of inclination b and the opposite side c, we can use the tangent ratio:
tangent(b) = opposite / adjacent
tangent(18°) = c / d
Next, we need to find the length of the adjacent side, which is the distance from the base of the tree to the top of the hillside. We can use the angle of elevation a and the tangent ratio:
tangent(a) = opposite / adjacent
tangent(48°) = h / d
Now we have two equations:
1. tangent(18°) = c / d
2. tangent(48°) = h / d
We can rearrange the first equation to solve for d:
d = c / tangent(18°)
Substitute this value of d into the second equation:
tangent(48°) = h / (c / tangent(18°))
Simplify and solve for h:
h = (c * tangent(48°)) / tangent(18°)
Now we can substitute the known values:
h = (235 * tangent(48°)) / tangent(18°)
Using a calculator, evaluate the tangent values for 48° and 18°, then substitute them into the equation to find h.
After calculating, the height of the tree is approximately 524 ft (rounded to the nearest whole number).