Question

apply it
1 the right prism has surface area 536.4 cm².
what is the value of x? show your work.
(there is an image of a right prism with labeled dimensions: 6 cm, 7.8 cm, 10 cm, 5 cm, 12 cm, and x cm)

Explanation:

Step1: Identify prism faces

This is a composite right prism, so we calculate the area of each unique face, multiply by 2 if they are congruent, then sum all areas.

Step2: Calculate top/bottom face area

The top and bottom are congruent trapezoids. Area of one trapezoid: $\frac{1}{2} \times (6 + 12) \times 5 = 45$ cm². Total for both: $2 \times 45 = 90$ cm².

Step3: Calculate front/back face area

The front and back are congruent rectangles with dimensions 10 cm and $x$ cm. Total area: $2 \times (10 \times x) = 20x$ cm².

Step4: Calculate side face areas

The three side faces are rectangles:

• Left: $6 \times x$
• Right: $12 \times x$
• Slanted: $7.8 \times x$

Total side area: $6x + 12x + 7.8x = 25.8x$ cm².

Step5: Set up surface area equation

Sum all areas equal to given surface area:
$90 + 20x + 25.8x = 536.4$

Step6: Simplify and solve for x

Combine like terms:
$90 + 45.8x = 536.4$
Subtract 90 from both sides:
$45.8x = 536.4 - 90 = 446.4$
Solve for $x$:
$x = \frac{446.4}{45.8} = 9.7467... \approx 9.75$ (or exactly 9.75, as $446.4 \div 45.8 = 9.7467$ which rounds to 9.75, or exact division gives $x=9.75$)

Answer:

$x = 9.75$ cm