Question

question 3 of 20. type the correct answer in each box. use numerals instead of words. the surface area of a rectangular prism is 288 square centimeters. the length of the prism is 3 times the height. the width is 2 centimeters longer than the height. what are the dimensions of the prism? length = cm, height = cm, width = cm

Explanation:

Step1: Let the height be $h$ cm.

Length $l = 3h$ cm, width $w=2$ cm. The surface - area formula for a rectangular prism is $S = 2(lw+lh + wh)$. Substitute $l = 3h$ and $w = 2$ into the formula: $S=2((3h\times2)+(3h\times h)+(2\times h))$.

Step2: Simplify the surface - area formula.

$S = 2(6h + 3h^{2}+2h)=2(3h^{2}+8h)=6h^{2}+16h$.

Step3: Set up the equation using the given surface area.

Since $S = 288$ square centimeters, we have the quadratic equation $6h^{2}+16h-288 = 0$. Divide the entire equation by 2 to simplify: $3h^{2}+8h - 144=0$.

Step4: Solve the quadratic equation.

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 3$, $b = 8$, $c=-144$), use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(8)^{2}-4\times3\times(-144)=64 + 1728 = 1792$. Then $h=\frac{-8\pm\sqrt{1792}}{6}=\frac{-8\pm\sqrt{256\times7}}{6}=\frac{-8\pm16\sqrt{7}}{6}=\frac{-4\pm8\sqrt{7}}{3}$. We take the positive root since height cannot be negative. $h=\frac{-4 + 8\sqrt{7}}{3}\approx\frac{-4+8\times2.646}{3}=\frac{-4 + 21.168}{3}=\frac{17.168}{3}\approx5.72$. But we can also factor the quadratic $3h^{2}+8h - 144 = 0$ as $(3h + 24)(h - 6)=0$. Solving for $h$, we get $h = 6$ or $h=-\frac{24}{3}=-8$. We discard the negative value.

Step5: Find the length and width.

If $h = 6$ cm, then $l=3h = 18$ cm and $w = 2$ cm.

Answer:

length = 18
height = 6
width = 2