Question
Solve. Give an exact answer and a two-decimal-place approximation where appropriate.
A wire is needed to support a vertical pole 10 feet high. The cable will be anchored to a stake 6 feet from the base of the pole. How much cable is needed?
10
6
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 10 feet high. The cable will be anchored to a stake 6 feet from the base of the pole. How much cable is needed?
10
6
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 find the length of cable needed to support the vertical pole, we can represent this situation using a right triangle where:
- The height of the pole is one leg of the triangle (10 feet).
- The distance from the base of the pole to the stake is the other leg of the triangle (6 feet).
- The length of the cable is the hypotenuse of the triangle.
We can use the Pythagorean theorem to find the length of the cable. The Pythagorean theorem states that:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two legs. In our case:
- \(a = 10\) feet (the height of the pole),
- \(b = 6\) feet (the distance from the base of the pole).
Now we can plug in the values:
\[
10^2 + 6^2 = c^2
\]
Calculating the squares:
\[
100 + 36 = c^2
\]
Adding these together:
\[
136 = c^2
\]
To find \(c\), we take the square root of both sides:
\[
c = \sqrt{136}
\]
We can simplify \(\sqrt{136}\):
\[
\sqrt{136} = \sqrt{4 \times 34} = \sqrt{4} \times \sqrt{34} = 2\sqrt{34}
\]
Thus, the exact length of the cable needed is:
\[
\text{Exact length} = 2\sqrt{34} \text{ feet}
\]
Now, to provide a two-decimal-place approximation, we calculate \(2\sqrt{34}\):
Calculating \(\sqrt{34} \approx 5.831\):
\[
2 \times 5.831 \approx 11.662
\]
Thus, rounded to two decimal places, the length of cable needed is approximately:
\[
\text{Approximate length} \approx 11.66 \text{ feet}
\]
In summary:
- **Exact length**: \(2\sqrt{34}\) feet
- **Approximate length**: \(11.66\) feet
- The height of the pole is one leg of the triangle (10 feet).
- The distance from the base of the pole to the stake is the other leg of the triangle (6 feet).
- The length of the cable is the hypotenuse of the triangle.
We can use the Pythagorean theorem to find the length of the cable. The Pythagorean theorem states that:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two legs. In our case:
- \(a = 10\) feet (the height of the pole),
- \(b = 6\) feet (the distance from the base of the pole).
Now we can plug in the values:
\[
10^2 + 6^2 = c^2
\]
Calculating the squares:
\[
100 + 36 = c^2
\]
Adding these together:
\[
136 = c^2
\]
To find \(c\), we take the square root of both sides:
\[
c = \sqrt{136}
\]
We can simplify \(\sqrt{136}\):
\[
\sqrt{136} = \sqrt{4 \times 34} = \sqrt{4} \times \sqrt{34} = 2\sqrt{34}
\]
Thus, the exact length of the cable needed is:
\[
\text{Exact length} = 2\sqrt{34} \text{ feet}
\]
Now, to provide a two-decimal-place approximation, we calculate \(2\sqrt{34}\):
Calculating \(\sqrt{34} \approx 5.831\):
\[
2 \times 5.831 \approx 11.662
\]
Thus, rounded to two decimal places, the length of cable needed is approximately:
\[
\text{Approximate length} \approx 11.66 \text{ feet}
\]
In summary:
- **Exact length**: \(2\sqrt{34}\) feet
- **Approximate length**: \(11.66\) feet