Question: 2. In a robotics competition, a robot must be built with a specific maximum starting height, width, and depth. In order to meet the height requirements, a robot must fit in the judge’s plexi-glass box. The box is a rectangular prism that has a surface area of 1 728 in², a width of 15 inches, and a length of 18 inches. The Mad Hatter robot is 17.75 inches tall in its starting position. Does the Mad Hatter meet the height requirement?
1. **State the problem:** We need to determine if the Mad Hatter robot, which is 17.75 inches tall, fits inside a plexi-glass box with given dimensions and surface area constraints.
2. **Given:**
- Surface area of box, $SA = 1728$ in²
- Width, $w = 15$ in
- Length, $l = 18$ in
- Height of robot, $h_{robot} = 17.75$ in
3. **Formula for surface area of a rectangular prism:**
$$SA = 2(lw + lh + wh)$$
4. **Find the height $h$ of the box:**
Substitute known values:
$$1728 = 2(18 \times 15 + 18 \times h + 15 \times h)$$
5. Simplify inside the parentheses:
$$1728 = 2(270 + 18h + 15h)$$
$$1728 = 2(270 + 33h)$$
6. Distribute 2:
$$1728 = 540 + 66h$$
7. Isolate $h$:
$$1728 - 540 = 66h$$
$$1188 = 66h$$
8. Divide both sides by 66:
$$h = \frac{1188}{66}$$
9. Simplify the fraction:
$$h = \frac{\cancel{1188}^{18} \times 66/18}{\cancel{66}^{1} \times 66/66} = 18$$
10. **Interpretation:** The height of the box is $18$ inches.
11. **Compare robot height to box height:**
Robot height = $17.75$ inches
Box height = $18$ inches
Since $17.75 < 18$, the Mad Hatter robot fits inside the box and meets the height requirement.