Scientist estimate the heights of features on the moon by measuring the lenghts of the shadows they cast on the moon's surface. From a photograph, you find that the shadow cast on the inside of a crater by its rim is 325 meters long. At the time the photograph was taken, the sun's angle of elevation from this place on the moon's surface was 23deg37minutes. How high does the rim rise above the inside of the crater?
Was helpful. 10 years ago response. Math concepts never change.
The only hard part I see about this problem is the way the angle is stated.
you will have to change it to
(23 + 37/60)° and then take the tangent ratio.
(on my calculator I have a key labelled D°M'S
so I could take
tan
23
D°M'S
37
=
and I get the same result as above.
BTW, did you get a height of 142.1 m ?
20.7
To determine the height of the rim of the crater above the inside, we need to use trigonometry and the concept of similar triangles. Here's how we can find the height:
1. Draw a diagram: Visualize the situation by drawing a diagram. Draw a right triangle representing the situation, with the vertical side representing the height we want to find, the horizontal side representing the distance from the rim to the base of the shadow, and the hypotenuse representing the length of the shadow.
2. Identify the known values: We are given the length of the shadow, which is 325 meters, and the angle of elevation of the sun, which is 23°37'.
3. Break down the angle of elevation: Convert the angle of elevation from degrees and minutes to decimal form. For example, 23°37' can be written as 23.62°.
4. Set up the trigonometric equation: We can use the tangent function to relate the angle of elevation to the height and the base of the shadow. The tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the opposite side is the height we want to find, and the adjacent side is the base of the shadow. The tangent equation is:
tan(angle) = opposite/adjacent
Let's call the height H and the base of the shadow B. The equation becomes:
tan(23.62°) = H/B
5. Solve the equation: Rearrange the equation to solve for H:
H = B * tan(23.62°)
Now we can substitute the value of B into the equation. However, we need to find the base of the shadow based on the angle of elevation and the length of the shadow.
6. Use trigonometry to find the base of the shadow: We can use the opposite function of the tangent, which is the arctangent function, to calculate the base of the shadow. The arctangent of an angle is equal to the opposite side divided by the adjacent side.
tan(angle) = opposite/adjacent
Rearrange the equation to solve for the adjacent side (base of the shadow):
adjacent = opposite/tan(angle)
Substituting the values:
adjacent = 325 meters / tan(23.62°)
7. Calculate the base of the shadow: Use a calculator to find the value of the adjacent side (base):
adjacent = 325 meters / tan(23.62°) ≈ 727.82 meters
8. Substitute the value of B into the equation to find H: Now we have the value of the base of the shadow, which is approximately 727.82 meters. Substitute this value into the equation we derived earlier:
H = 727.82 meters * tan(23.62°)
Calculate this equation using a calculator:
H ≈ 312.75 meters
Therefore, the rim of the crater rises approximately 312.75 meters above the inside.