Question

Suppose a spider was able to create one thread that would extend from the top-right back corner of its tank to the bottom-left front corner. Approximately how long would that thread need to be for the tank below? The path of the spider's thread is shown with a red dotted line. Round your answer to the nearest tenth of an inch.



35 in
16 in
12 in

Answer 1

To find the length of the thread the spider creates from the top-right back corner to the bottom-left front corner of the tank, we can visualize the tank as a cuboid with dimensions:

- Width (front to back): 35 inches
- Height (bottom to top): 16 inches
- Depth (left to right): 12 inches

The thread forms the hypotenuse of a right triangle in a three-dimensional space. We can use the 3D distance formula to find the length of the thread, which is expressed as follows:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

In this scenario:
- From \( (0, 0, 0) \) (bottom-left front corner) to \( (35, 16, 12) \) (top-right back corner):
- \( x_1 = 0 \), \( y_1 = 0 \), \( z_1 = 0 \)
- \( x_2 = 35 \), \( y_2 = 16 \), \( z_2 = 12 \)

Now, substituting these values into the formula:

\[
d = \sqrt{(35 - 0)^2 + (16 - 0)^2 + (12 - 0)^2}
\]

Calculating each squared term:

\[
d = \sqrt{35^2 + 16^2 + 12^2}
\]
\[
= \sqrt{1225 + 256 + 144}
\]
\[
= \sqrt{1625}
\]

Now we calculate \( \sqrt{1625} \):

\[
\sqrt{1625} \approx 40.311
\]

Rounding to the nearest tenth gives us:

\[
\approx 40.3 \text{ inches}
\]

Thus, the approximate length of the spider's thread is **40.3 inches**.