A ramp with a 30° angle from the ground is to be built up to a 2-foot high platform. What will be the length of the ramp? How far from the platform will it extend to the nearest inch?
With problems like these, it’s very helpful to draw a picture. So, we have a ramp that makes a 30° angle with the ground. So, let’s start with that.
The transcript is provided for your convenience
Here’s the ground, here’s our 30° angle, and it’s to be built up to a 2-foot high platform, so then, here we have a platform, and it’s 2 feet high above the ground. There’s a right angle there, perpendicular to the ground.
The first question is, what will be the length of the ramp? So, this is what we’re trying to find, but before I start to find x, I first want to fill in what this angle is.
You might already know that since we have a 30° angle and a 90° angle, that this angle has to be 60°, but if you didn’t know that, you could find it because there are 180° in a triangle, so if you add 30 to 90, you get 120, 180 minus 120, we use the 60° for that third angle.
Now that we know it’s a 30-60-90 triangle, we can apply our 30-60-90 rules to finding the length of our ramp. Our ramp is across from the 90° angle, therefore, that’s our hypotenuse.
The side we know, the leg we know is across from the 30° angle, so it’s our shorter leg. And the hypotenuse of the 30-60-90 triangle is twice as long as the shorter leg. Our hypotenuse, again, we’re using x for, equals 2 times the shorter leg, which is that 2-foot side, so 2 times 2 feet, which means x is 4 feet. Therefore, the length of the ramp is 4 feet.
So, we’ve answered the first question. Then they asked how far from the platform will it extend? How far from the platform will the ramp extend? So, now, they’re asking us to find that length. And y is across from our 60° angle, so that’s our longer leg, and the longer leg of a 30-60-90 triangle is the square root of 3, times the shorter leg. Our longer leg, again, is y, equals the square root of 3, times, and our shorter leg is still that 2-foot length. So, y is 2 square roots of 3 feet.
They did not ask us for the answer in feet, they asked to the nearest inch. So, the first thing we’re going to do is convert 2 square roots of 3 feet into inches by multiplying by 12. So, y is 2 square roots of 3 times 12, since there are 12 inches in 1 foot, which is 24 square roots of 3 inches.
But, they did not ask for it in simplest radical form, so then, we’d actually want to calculate what 24 square roots of 3 inches is, and it’s about 42 inches. Therefore, how far from the platform will the ramp extend? The ramp will extend about 42 inches from the platform. There we have it.