For an average​ person, the rate of change of weight W​ (in pounds) with respect to height h​ (in inches) is given approximately by the following formula.dW/dh=0.0018h^2Find​ W(h) if ​W(80​)equals=287.2 pounds. ​ Also, find the weight of a person who is 5​ feet, 8 inches tall.W(h)=_______A Person who is 5 feet, 8 inches tall weighs about _______ lbs

Answer:[tex]W(h)=0.0006h^3-20[/tex].A Person who is 5 feet, 8 inches tall weighs about 168.7 lbs.Step-by-step explanation:We know the rate of change of weight W​ (in pounds) with respect to height h​ (in inches) [tex]\frac{dW}{dh}=0.0018h^2[/tex]This is a separable equation. A separable equation is a first-order differential equation in which the expression for [tex]\frac{dy}{dx}[/tex] can be factored as a function of x times a function of y. In other words, it can be written in the form[tex]\frac{dy}{dx}=g(x)f(y)[/tex]To find W(h), we write the equation in terms of differentials and integrate both sides:[tex]\frac{dW}{dh}=0.0018h^2\\\\dW=(0.0018h^2)dh\\\\\int dW=\int (0.0018h^2)dh\\\\W=0.0006h^3+C[/tex]To find the value of C, we use W(80​) = 287.2 lbs[tex]287.2=0.0006(80)^3+C\\0.0006\left(80\right)^3+C=287.2\\307.2+C=287.2\\307.2+C-307.2=287.2-307.2\\C=-20[/tex]Thus, [tex]W(h)=0.0006h^3-20[/tex]To find the weight of a person who is 5​ feet, 8 inches tall you must:Convert the 5 feet into inches[tex]5 \:ft \:\frac{12 \:in}{1\:ft} = 60 \:in[/tex]Add 60 in and 8 in, to find the total height of the personh = 68 inSubstitute h = 68 in into [tex]W(h)=0.0006h^3-20[/tex] to find the weight:[tex]W(68)=0.0006(68)^3-20=168.7[/tex]A Person who is 5 feet, 8 inches tall weighs about 168.7 lbs.