A tree is growing at the edge of a cliff. From the tree, the angle between the base of the cliff and the base of a nearby house is 62 degrees. If the distance between the base of the cliff and the base of the house is 500 feet, how many feet tall is the cliff?
Assuming the cliff is vertical and of height h feet, we have
500/h = tan 62° = 1.88
h = 265.85 or 266
To solve this problem, we can use trigonometry.
1. Draw a diagram to better understand the situation. Let's label the base of the cliff as "A," the base of the house as "B," and the top of the tree as "C."
2. Given that the angle between the base of the cliff (A) and the base of the house (B) is 62 degrees, we can label this angle as θ.
3. Use the tangent function to find the height of the cliff (AC). The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the cliff (AC), and the adjacent side is the distance between the base of the cliff and the base of the house (AB).
tan(θ) = AC / AB
4. Substitute the known values into the equation:
tan(62 degrees) = AC / 500 feet
5. Solve for AC by multiplying both sides of the equation by 500 feet:
AC = tan(62 degrees) * 500 feet
6. Calculate the value of tan(62 degrees):
tan(62 degrees) ≈ 1.880726465
(You can use a scientific calculator or an online calculator to find the tan of 62 degrees.)
7. Multiply the value of tan(62 degrees) by 500 feet to find the height of the cliff:
AC ≈ 1.880726465 * 500 feet
8. Calculate AC:
AC ≈ 940.36 feet
Therefore, the height of the cliff is approximately 940.36 feet.
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
In this problem, if we consider the tree as the point of reference, the angle between the base of the cliff and the base of the house is given as 62 degrees. We can call this angle θ.
Let's label the height of the cliff as 'h'.
Using the tangent function, we have:
tan(θ) = opposite / adjacent
Applying this to the given information, we get:
tan(62°) = h / 500
Now, we need to find the value of tan(62°) to determine the height of the cliff.
To find the value of tan(62°), you can use a scientific calculator or an online calculator that supports trigonometric functions. Simply input 62° into the calculator, press the tangent function or the "tan" button, and it will give you the value of tan(62°).
Once you have the value of tan(62°), substitute it into the equation:
tan(62°) = h / 500
Now, we can solve for 'h' by rearranging the equation:
h = tan(62°) * 500
Finally, calculate the value of h using the calculated value of tan(62°) and multiply it by 500. The resulting value will give you the height of the cliff in feet.