a waste management company is designing a rectangular construction dumpster that will be twice as long as it…
a waste management company is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold 25 yd³ of debris. find the dimensions of the dumpster that will minimize its surface area. write the surface area formula in terms of the width, x. assume the dumpster has an open top. sa = □
Answer
Explanation:
Step1: Determine the length
Given the width is (x), and the length is twice the width, so the length (l = 2x).
Step2: Express the height in terms of (x)
The volume (V=l\times w\times h), and (V = 25) yd³. Substituting (l = 2x) and (w=x) into the volume formula (V=2x\cdot x\cdot h=2x^{2}h). Then (h=\frac{25}{2x^{2}}).
Step3: Write the surface - area formula
The surface - area formula for an open - top rectangular box is (SA=lw + 2lh+2wh). Substitute (l = 2x), (w = x), and (h=\frac{25}{2x^{2}}) into the formula: [ \begin{align*} SA&=(2x\cdot x)+2(2x)\cdot\frac{25}{2x^{2}}+2(x)\cdot\frac{25}{2x^{2}}\ &=2x^{2}+\frac{50x}{2x^{2}}+\frac{25x}{x^{2}}\ &=2x^{2}+\frac{25}{x}+\frac{25}{x}\ &=2x^{2}+\frac{50}{x} \end{align*} ]
Answer:
(SA = 2x^{2}+\frac{50}{x})