Question
Solve. Give an exact answer and a two-decimal-place approximation where appropriate.
A wire is needed to support a vertical pole 20 feet high. The cable will be anchored to a stake 9 feet from the base of the pole. How much cable is needed?
20
9
Question content area bottom
Part 1
The exact length is
enter your response here feet.
(Type an exact answer, using radicals as needed. Simplify your answer.)
A wire is needed to support a vertical pole 20 feet high. The cable will be anchored to a stake 9 feet from the base of the pole. How much cable is needed?
20
9
Question content area bottom
Part 1
The exact length is
enter your response here feet.
(Type an exact answer, using radicals as needed. Simplify your answer.)
Answers
GPT-4o mini
To determine the length of the cable needed to support the vertical pole, we can use the Pythagorean Theorem. The scenario forms a right triangle where:
- One leg is the height of the pole (20 feet),
- The other leg is the distance from the base of the pole to the stake (9 feet),
- The hypotenuse is the length of the cable we need to find.
The Pythagorean Theorem states that:
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is one leg of the triangle (20 feet),
- \( b \) is the other leg (9 feet),
- \( c \) is the hypotenuse (the length of the cable).
Substituting the values we have:
\[
20^2 + 9^2 = c^2
\]
Calculating the squares:
\[
400 + 81 = c^2
\]
\[
481 = c^2
\]
Now, we solve for \( c \) by taking the square root:
\[
c = \sqrt{481}
\]
This can be simplified further if possible. The prime factorization of 481 shows it does not have any perfect square factors (it is actually \( 13 \times 37 \)), so we leave it as is:
\[
c = \sqrt{481}
\]
### Exact Length
Thus, the exact length of cable needed is:
\[
\sqrt{481} \text{ feet}
\]
### Decimal Approximation
Now, let's compute the decimal approximation:
\[
\sqrt{481} \approx 21.93
\]
So, the two-decimal-place approximation of the length of the cable is:
\[
21.93 \text{ feet}
\]
### Final Answers
Exact length: \(\sqrt{481} \, \text{feet}\)
Approximate length: \(21.93 \, \text{feet}\)
- One leg is the height of the pole (20 feet),
- The other leg is the distance from the base of the pole to the stake (9 feet),
- The hypotenuse is the length of the cable we need to find.
The Pythagorean Theorem states that:
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is one leg of the triangle (20 feet),
- \( b \) is the other leg (9 feet),
- \( c \) is the hypotenuse (the length of the cable).
Substituting the values we have:
\[
20^2 + 9^2 = c^2
\]
Calculating the squares:
\[
400 + 81 = c^2
\]
\[
481 = c^2
\]
Now, we solve for \( c \) by taking the square root:
\[
c = \sqrt{481}
\]
This can be simplified further if possible. The prime factorization of 481 shows it does not have any perfect square factors (it is actually \( 13 \times 37 \)), so we leave it as is:
\[
c = \sqrt{481}
\]
### Exact Length
Thus, the exact length of cable needed is:
\[
\sqrt{481} \text{ feet}
\]
### Decimal Approximation
Now, let's compute the decimal approximation:
\[
\sqrt{481} \approx 21.93
\]
So, the two-decimal-place approximation of the length of the cable is:
\[
21.93 \text{ feet}
\]
### Final Answers
Exact length: \(\sqrt{481} \, \text{feet}\)
Approximate length: \(21.93 \, \text{feet}\)