Question
Jude is making a door for a door frame with an opening that measures 9 feet by 3 feet. Which diagonal length should he use to make a door that fits in the door frame?
10 feet
3 StartRoot 10 EndRoot feet
6 StartRoot 2 EndRoot feet
12 feet
10 feet
3 StartRoot 10 EndRoot feet
6 StartRoot 2 EndRoot feet
12 feet
Answers
GPT-4o mini
To determine the diagonal length of the door that Jude needs to fit into the door frame, we can use the Pythagorean theorem. The diagonal \(d\) of a rectangle can be found using the formula:
\[
d = \sqrt{w^2 + h^2}
\]
where \(w\) is the width and \(h\) is the height of the rectangle.
In this case, the door frame has an opening measuring 9 feet (height) by 3 feet (width). Therefore, we can plug these dimensions into the formula:
\[
d = \sqrt{(3)^2 + (9)^2}
\]
\[
d = \sqrt{9 + 81}
\]
\[
d = \sqrt{90}
\]
Now we can simplify \(\sqrt{90}\):
\[
\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}
\]
Now, let's compare \(3\sqrt{10}\) with the provided options to find which one matches:
1. 10 feet
2. \(3\sqrt{10}\) feet
3. \(6\sqrt{2}\) feet
4. 12 feet
The diagonal length that Jude should use to make a door that fits in the door frame is **\(3\sqrt{10}\) feet**.
\[
d = \sqrt{w^2 + h^2}
\]
where \(w\) is the width and \(h\) is the height of the rectangle.
In this case, the door frame has an opening measuring 9 feet (height) by 3 feet (width). Therefore, we can plug these dimensions into the formula:
\[
d = \sqrt{(3)^2 + (9)^2}
\]
\[
d = \sqrt{9 + 81}
\]
\[
d = \sqrt{90}
\]
Now we can simplify \(\sqrt{90}\):
\[
\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}
\]
Now, let's compare \(3\sqrt{10}\) with the provided options to find which one matches:
1. 10 feet
2. \(3\sqrt{10}\) feet
3. \(6\sqrt{2}\) feet
4. 12 feet
The diagonal length that Jude should use to make a door that fits in the door frame is **\(3\sqrt{10}\) feet**.