Question
Ana recycles a cookie tin to use for storing school supplies. The tin is a rectangular prism with a length of 14 centimeters, a width of 10 centimeters, and a height of 4 centimeters.
What is the length of the longest pencil that can fit inside the tin? Enter the answer in the box. Round the answer to the nearest hundredth of a centimeter
What is the length of the longest pencil that can fit inside the tin? Enter the answer in the box. Round the answer to the nearest hundredth of a centimeter
Answers
GPT-4o mini
To find the length of the longest pencil that can fit inside the rectangular prism (cookie tin), we need to calculate the space diagonal of the prism. The formula for the space diagonal \( d \) of a rectangular prism with dimensions \( l \), \( w \), and \( h \) is given by:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
Where:
- \( l \) is the length of the prism (14 cm)
- \( w \) is the width of the prism (10 cm)
- \( h \) is the height of the prism (4 cm)
Plugging in the values:
\[
d = \sqrt{14^2 + 10^2 + 4^2}
\]
Calculating each term:
\[
14^2 = 196
\]
\[
10^2 = 100
\]
\[
4^2 = 16
\]
Now, adding these values:
\[
d = \sqrt{196 + 100 + 16} = \sqrt{312}
\]
Now, finding the square root of 312:
\[
d \approx 17.577
\]
Finally, rounding this to the nearest hundredth:
\[
d \approx 17.58
\]
Thus, the length of the longest pencil that can fit inside the tin is
\[
\boxed{17.58} \text{ cm}
\]
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
Where:
- \( l \) is the length of the prism (14 cm)
- \( w \) is the width of the prism (10 cm)
- \( h \) is the height of the prism (4 cm)
Plugging in the values:
\[
d = \sqrt{14^2 + 10^2 + 4^2}
\]
Calculating each term:
\[
14^2 = 196
\]
\[
10^2 = 100
\]
\[
4^2 = 16
\]
Now, adding these values:
\[
d = \sqrt{196 + 100 + 16} = \sqrt{312}
\]
Now, finding the square root of 312:
\[
d \approx 17.577
\]
Finally, rounding this to the nearest hundredth:
\[
d \approx 17.58
\]
Thus, the length of the longest pencil that can fit inside the tin is
\[
\boxed{17.58} \text{ cm}
\]