The ratio of horizontal distance to height of the ramp is 23:2. A builder has a roll of non-slip rubber mat that is 23 feet long. Does he have enough rubber to cover the ramp completely? Explain.
Select the correct choice below, and fill in the answer box to complete your choice. (Simplify your answer. Type an exact answer, using radicals as needed.)
A. Yes. The length of the ramp is _____ feet. The rubber mat is equal to this length.
B. Yes. The length of the ramp is _____ feet. The rubber mat is longer than this length.
C. No. The length of the ramp is _____ feet. The rubber mat will be too short.
To determine if the rubber mat is enough to cover the ramp, we need to find the length of the ramp using the given ratio of horizontal distance to height, which is 23:2.
1. Let the horizontal distance be (23x) and the height be (2x).
2. We can use the Pythagorean theorem to find the length of the ramp (hypotenuse):
[
text{Length of ramp} = sqrt{(23x)^2 + (2x)^2}
]
3. Simplifying this:
[
= sqrt{529x^2 + 4x^2}
]
[
= sqrt{533x^2}
]
[
= xsqrt{533}
]
4. Now, we need to check how long the ramp can be with the rubber mat of 23 feet. Set the length of the ramp equal to 23:
[
xsqrt{533} leq 23
]
[
x leq frac{23}{sqrt{533}}
]
5. However, let’s ascertain the numerical value of ( sqrt{533} ):
[
sqrt{533} approx 23.065
]
Thus,
[
x leq frac{23}{23