35. Suppose that the volume, V(x), of a box with a square base and an open top satisfies the function below. Find a positive critical value for the function V. V (x) = (1200x – x³) 30 20 5 24
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- A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in inches. a. Using the fact that the volume of the can is 25 cubic inches, express h in terms of x. b. Express the total surface area S of the can in terms of x.Find the critical point of ƒ(x, y) = xy + 2x - ln x2y in the open first quadrant (x >0, y>0) and show that ƒ takes on a minimum there.In the function: f(x)= (3x^2)ln(x) , x>0 What are the critical numbers of the function?
- A manufacturer incurs the following costs in producing x water ski vests in one day, for 0<x<150: fixed costs, $500; unit production cost, $25 per vest; equipment maintenance and repairs, 0.05x2 dollars. So, the cost of manufacturing x vests in one given day is given by C(x)=0.05x2+25x+500,where 0<x<150. a.) What is the average cost C(x) per vest if x vests are produced in one day? b.) Find the critical numbers of C(x), the intervals on which the average cost per vest is decreasing, the intervals on which the average cost per vest is increasing, and the local extrema.A manufacturer incurs the following costs in producing x water ski vests in one day, for 0<x<200: fixed costs, $245; unit production cost, $15 per vest; equipment maintenance and repairs, 0.05x2 dollars. So, the cost of manufacturing x vests in one given day is given by C(x)=0.05x2+15x+245, where 0<x<200. (A) What is the average cost C(x) per vest if x vests are produced in one day? (B) Find the critical numbers of C(x), the intervals on which the average cost per vest is decreasing, the intervals on which the average cost per vest is increasing, and the local extrema. Do not graphFind the center of mass of the thin plate bounded by the graphs of the given functions. g(x) = x^2(x+ 1), f(x) = 2, and x= 0.
- Suppose that X and Y are independent measurement of a quantity μ. E(X) = E(Y) = μ, σX ≠ σY, Z = αX + (1 - α)Y, α ∈ [0, 1] (a) Show that E(X) = μ. (b) Find α in terms of σX and σY to minimize V(Z). (c) Under what circumstance is it better to use the average (X+Y)/2 than either X or Y alone?find the minimum and maximum value of the function on. the given interval by comparing values at the critical points and endpoints. y = −x , [0, 2] ln x, [1, 3]A population of E. coli bacteria will grow at a rate given by w'(t)=(3t+2)^1/3 where w is the weight in mg after t hours If the region is enclosed by y=x^2 and y=2x, estimate the area by the right endpoints with 4 subintervals