Question
Oscar's dog house is shaped like a tent. The slanted sides are both 5 feet long and the bottom of the house is 6 feet across. What is the height of his dog house, in feet, at its tallest point? Round to the nearest tenth.
Question 9 options:
4
8.1
3
5.8
Question 9 options:
4
8.1
3
5.8
Answers
GPT-4o mini
To find the height of Oscar's dog house, we can model it as a triangle where the base is the width of the dog house and the slanted sides are the lengths of the tent.
The base of the triangle is 6 feet. Since the tent shape is symmetrical, the height can be found by dividing the base into two equal halves, making two right triangles. Therefore, each half of the base is:
\[
\frac{6}{2} = 3 \text{ feet}
\]
Now we have a right triangle where:
- One leg (half of the base) is 3 feet.
- The hypotenuse (the slanted side) is 5 feet.
- The other leg (the height of the dog house) is what we are trying to find.
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \(a\) is the height (unknown), \(b\) is 3 feet, and \(c\) is 5 feet, we can write:
\[
a^2 + 3^2 = 5^2
\]
Calculating the squares:
\[
a^2 + 9 = 25
\]
Now, isolate \(a^2\):
\[
a^2 = 25 - 9 = 16
\]
Taking the square root of both sides gives:
\[
a = \sqrt{16} = 4 \text{ feet}
\]
Thus, the height of Oscar's dog house, at its tallest point, is:
\[
\boxed{4} \text{ feet}
\]
The base of the triangle is 6 feet. Since the tent shape is symmetrical, the height can be found by dividing the base into two equal halves, making two right triangles. Therefore, each half of the base is:
\[
\frac{6}{2} = 3 \text{ feet}
\]
Now we have a right triangle where:
- One leg (half of the base) is 3 feet.
- The hypotenuse (the slanted side) is 5 feet.
- The other leg (the height of the dog house) is what we are trying to find.
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where \(a\) is the height (unknown), \(b\) is 3 feet, and \(c\) is 5 feet, we can write:
\[
a^2 + 3^2 = 5^2
\]
Calculating the squares:
\[
a^2 + 9 = 25
\]
Now, isolate \(a^2\):
\[
a^2 = 25 - 9 = 16
\]
Taking the square root of both sides gives:
\[
a = \sqrt{16} = 4 \text{ feet}
\]
Thus, the height of Oscar's dog house, at its tallest point, is:
\[
\boxed{4} \text{ feet}
\]