Ask a New Question
Menu

What is the hypotenuse (c) in this question? Also, if a=5 feet, which side do we need to find?

Wendy and Lilly were playing ball in the yard when the ball got stuck in the tree. It is too high for them to reach, so they need to get a ladder. When the 13 ft. ladder is leaned against the tree, the top of the ladder is even with the ball. The distance between the base of the tree and the bottom of the ladder is 5 ft. How high is the ball in the tree?

11 answers

To find the height of the ball in the tree, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).

In this scenario:

  • The length of the ladder is the hypotenuse (c) = 13 feet.
  • The distance from the base of the tree to the bottom of the ladder (one leg of the triangle) is a = 5 feet.
  • We need to find the height of the ball in the tree, which will be the other leg of the triangle (b).

Using the Pythagorean theorem formula:

\[ c^2 = a^2 + b^2 \]

Substituting the known values into the formula:

\[ 13^2 = 5^2 + b^2 \]

Calculating the squares:

\[ 169 = 25 + b^2 \]

Then, isolate \(b^2\):

\[ b^2 = 169 - 25 \] \[ b^2 = 144 \]

Now, take the square root of both sides to find \(b\):

\[ b = \sqrt{144} = 12 \]

So, the height of the ball in the tree is 12 feet.

what is the hypotenuse and what side are we looking for? a b or c?

In the problem you provided, the hypotenuse is represented by \(c\), and it is the length of the ladder, which is 13 feet.

You are looking for side \(b\), which represents the height of the ball in the tree. The lengths are defined as follows:

  • \(a\) = distance from the base of the tree to the bottom of the ladder = 5 feet (one leg of the triangle).
  • \(b\) = height of the ball in the tree = 12 feet (the other leg of the triangle, which we calculated).
  • \(c\) = length of the ladder = 13 feet (the hypotenuse).

So, to summarize:

  • The hypotenuse \(c\) is 13 feet.
  • You are looking for side (b**, which is the height of the ball, calculated to be 12 feet.
And what formula can we use to find how high the ball is in the tree?

To find how high the ball is in the tree, you can use the Pythagorean theorem. The Pythagorean theorem relates the lengths of the sides of a right triangle and is expressed by the formula:

\[ c^2 = a^2 + b^2 \]

Where:

  • \(c\) is the length of the hypotenuse (the ladder in this case),
  • \(a\) is one leg of the triangle (the distance from the base of the tree to the bottom of the ladder),
  • \(b\) is the other leg of the triangle (the height of the ball in the tree, which we are trying to find).

In your scenario, here’s how to use the formula:

  1. Identify the known values:

    • \(c = 13\) feet (length of the ladder)
    • \(a = 5\) feet (distance from the base of the tree to the bottom of the ladder)
    • \(b\) is the unknown we want to find (the height of the ball).

  2. Set up the equation using the Pythagorean theorem:

\[ 13^2 = 5^2 + b^2 \]

  1. Calculate the squares:
  • \(13^2 = 169\)
  • \(5^2 = 25\)
  1. Substitute and solve for \(b^2\):

\[ 169 = 25 + b^2 \] \[ b^2 = 169 - 25 \] \[ b^2 = 144 \]

  1. Take the square root to find \(b\):

\[ b = \sqrt{144} = 12 \]

So, using the Pythagorean theorem, you can find the height of the ball in the tree, which is 12 feet.

you sure?

Yes, I'm sure. The use of the Pythagorean theorem is accurate in this context, given that we have a right triangle formed by the ladder, the ground, and the height of the tree where the ball is stuck.

To summarize the key points step-by-step:

  1. Identify the triangle's sides:

    • The ladder, which acts as the hypotenuse \(c\), is 13 feet long.
    • The distance from the base of the tree to the bottom of the ladder \(a\) is 5 feet.
    • We need to find the height of the ball in the tree, which we will call \(b\).

  2. Apply the Pythagorean theorem:

    The formula is:

    \[ c^2 = a^2 + b^2 \]

    Substituting the known values:

    \[ 13^2 = 5^2 + b^2 \]

  3. Calculate:

    • \(13^2 = 169\)
    • \(5^2 = 25\)

    Plugging these back into the equation gives:

    \[ 169 = 25 + b^2 \]

    Solving for \(b^2\):

    \[ b^2 = 169 - 25 \] \[ b^2 = 144 \]

  4. Solve for \(b\):

    Taking the square root:

    \[ b = \sqrt{144} = 12 \]

So, the height of the ball in the tree is correctly calculated to be 12 feet.

If you have any further questions or if something seems unclear, please feel free to ask!

you sure

Yes, I am confident in the calculations and the conclusion. Here’s a concise recap of the process to clarify:

  1. Understanding the situation:

    • You have a right triangle where:
      • The ladder acts as the hypotenuse (\(c\)) = 13 feet.
      • The ground distance from the bottom of the ladder to the base of the tree (\(a\)) = 5 feet.
      • The height of the ball in the tree (\(b\)) is the side we need to find.

  2. Using the Pythagorean theorem: The theorem states: \[ c^2 = a^2 + b^2 \]

  3. Plugging in the values: \[ 13^2 = 5^2 + b^2 \] Which simplifies to: \[ 169 = 25 + b^2 \]

  4. Isolating \(b^2\): \[ b^2 = 169 - 25 \] \[ b^2 = 144 \]

  5. Finding \(b\): \[ b = \sqrt{144} = 12 \]

So the height of the ball in the tree is indeed 12 feet.

I understand if there may still be doubts; please let me know if you have any specific questions or if there are any other aspects of the problem you'd like to discuss!

What is the formula we can use to find how high the ball is in the tree?

To find how high the ball is in the tree, you can use the Pythagorean theorem, which is appropriate for right triangles. The Pythagorean theorem states:

\[ c^2 = a^2 + b^2 \]

Where:

  • \(c\) is the length of the hypotenuse (the ladder in this situation).
  • \(a\) is one leg of the triangle (the horizontal distance from the base of the tree to the bottom of the ladder).
  • \(b\) is the other leg of the triangle (the height of the ball in the tree, which you want to find).

Steps to Use the Pythagorean Theorem:

  1. Identify the values:

    • \(c = 13\) feet (length of the ladder).
    • \(a = 5\) feet (distance from the base of the tree to the ladder).

  2. Set up the equation using the Pythagorean theorem:

    \[ 13^2 = 5^2 + b^2 \]

  3. Calculate:

    \[ 169 = 25 + b^2 \]

  4. Isolate \(b^2\):

    \[ b^2 = 169 - 25 \] \[ b^2 = 144 \]

  5. Take the square root to find \(b\):

    \[ b = \sqrt{144} = 12 \]

Conclusion:

The height of the ball in the tree is 12 feet.

So, the formula you need is the Pythagorean theorem, \(c^2 = a^2 + b^2\), to determine the height of the ball. If you have any further questions or if you'd like clarification on any part of this, feel free to ask!