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. Show your work. The length is 35 in, 16 in high and 12 in in width
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. Show your work.

Answer 1

To find the length of the thread that the spider would create from the top-right back corner of the tank to the bottom-left front corner, we can use the three-dimensional distance formula. The tank's dimensions are given as follows:

- Length (width) = 35 inches
- Height = 16 inches
- Depth = 12 inches

We will define the coordinates of the points we are interested in.

1. **Top-right back corner** of the tank:
This corner can be defined as (35, 0, 16) where:
- 35 is the length (x-coordinate),
- 0 is the depth (y-coordinate),
- 16 is the height (z-coordinate).

2. **Bottom-left front corner** of the tank:
This corner can be defined as (0, 12, 0) where:
- 0 is the length (x-coordinate),
- 12 is the depth (y-coordinate),
- 0 is the height (z-coordinate).

### Using the Distance Formula

The distance \( d \) between two points in three-dimensional space \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by the formula:

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

### Apply the Coordinates

Plugging in the coordinates from above:
- \( (x_1, y_1, z_1) = (35, 0, 16) \)
- \( (x_2, y_2, z_2) = (0, 12, 0) \)

The formula becomes:

\[
d = \sqrt{(0 - 35)^2 + (12 - 0)^2 + (0 - 16)^2}
\]
\[
= \sqrt{(-35)^2 + (12)^2 + (-16)^2}
\]
\[
= \sqrt{1225 + 144 + 256}
\]
\[
= \sqrt{1625}
\]

### Calculating the Square Root

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

\[
\sqrt{1625} \approx 40.31
\]

### Rounding

Finally, rounding \( 40.31 \) to the nearest tenth gives us:

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

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