2 questions:
An airplane takes off from the ground and reaches a height of 500 feet after flying 2 miles. Given the formula H=dtan times theta, where H is the height of the plane and d is the distance (along the ground) the plane has flown, find the angle of ascent theta at which the plane took off.
H = d tan0
tan0 = H/d = 500 ft/10,560 ft = .04735
0 = 2.71 degrees
"0" was supposed to be the symbol theta in my previous post
I want to expand drwls' reply.
First, convert 2 miles to feet.
If 1 mile = 5280, then 2 miles = 10,560 feet.
Secondly, you must re-arrange the given formula because we are looking for an angle NOT height or distance on ground.
Let t = theta FOR SHORT.
tan(t) = 500ft/2 miles
tan(t) = 500ft/10,560ft
tan(t) = 0.047348485
Since we are looking an an angle, you must use the tangent invere key on your calculator. By the way, you must ALWAYS use the inverse key when searching for a trig function. I will do this in just a minute.
Before dealing with the angle itself, the decimal number 0.047348485 must be rounded to the nearest hundred- thousandths; it becomes 0.04735.
We now have:
tan(t) = 0.04735 We take the tangent inverse of the right side of the equation.
NOTE: tan^-1 is read: "tangent inverse" NOT tangent to the negative one. Is this clear?
t = tan^-1(0.04735)
t = 2.71 degrees
NOTE: I rounded 2.710843763 to the nearest hundreths and the answer became 2.71 degrees. This is what drwls did without going into detail.
I think detail is vital when tutoring online.
Done!
To find the angle of ascent theta at which the plane took off, we can rearrange the given formula H = d * tan(theta) to solve for theta.
1. Start with the formula H = d * tan(theta).
2. Substitute the given values: H = 500 feet and d = 2 miles.
Note: Since the units for H and d are different, we need to convert them to the same unit. Let's convert miles to feet, since H is given in feet.
1 mile = 5,280 feet, so 2 miles = 2 * 5,280 = 10,560 feet.
Therefore, H = 10,560 feet and d = 10,560 feet.
3. Now, the formula becomes 10,560 = 10,560 * tan(theta).
4. Divide both sides of the equation by 10,560: 10,560 / 10,560 = 10,560 * tan(theta) / 10,560.
This simplifies to 1 = tan(theta).
5. To solve for theta, we take the inverse tangent (also known as arctan) of both sides of the equation: theta = arctan(1).
6. Use a calculator or reference table to find the inverse tangent of 1. In degrees, arctan(1) is equal to 45 degrees.
Therefore, the angle of ascent theta at which the plane took off is 45 degrees.