Markus's kite drops so that the angle of elevation is 48 degrees. Find the height of the kite above the ground. the hypotenuse is 40 yds.
Sketch it out : )
You see that the angle of elevation is from the ground looking up at the kite. The angle of elevation is between the arm of the triangle that is on the ground and the hypotenuse.
You need the value of the OPPOSITE side of the triangle, that is the height.
So you have the HYPOTENUSE and need the OPPOSITE.
Which trig ratio deals with these two sides? SOH CAH TOA
sin48 = h/40.
To find the height of the kite above the ground, we can use trigonometric ratios. In this case, we'll use the tangent function.
1. Recall that the tangent of an angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
2. Let's denote the height of the kite as "h" and the length of the adjacent side as "x".
3. We know that the angle of elevation is 48 degrees, so the tangent of 48 degrees is equal to h/x. Thus, we have the equation tan(48°) = h/x.
4. The hypotenuse is given as 40 yards, and we can use this information to find the value of x using the Pythagorean theorem: (adjacent side)^2 + (opposite side)^2 = (hypotenuse)^2.
Plugging in the values, we have:
x^2 + h^2 = 40^2.
5. Rearrange the equation to solve for x^2: x^2 = 40^2 - h^2.
6. Replace x in the equation from step 3 with the value from step 5: (40^2 - h^2) / tan(48°) = h.
7. Simplify the equation and solve for h:
Subtract h^2 from both sides: 40^2 - h^2 = h * tan(48°).
Expand: 1600 - h^2 = h * tan(48°).
Rearrange the equation: h * tan(48°) + h^2 = 1600.
Factor out h: h * (tan(48°) + h) = 1600.
Divide both sides by (tan(48°) + h): h = 1600 / (tan(48°) + h).
Now we have an equation with h on both sides. To solve it, we'll need to use numerical methods or approximation techniques.
8. Substitute the value of tan(48°) into the equation and calculate h using a calculator or computer software.
By following this process, you can find the height of the kite above the ground.