Calculus Answers

Find the area, take the elements of the area perpendicular to the x-axis. x²-y+1=0; x-y+1=0.

A farmer wants to fence an area of 60,000 m² in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be (in m) in order to minimize the cost of the fence?

smaller value = m

larger value = m

A poster is to have an area of 630 cm² with 2.5 cm margins at the bottom and sides and a 5 cm margin at the top. Find the exact dimensions (in cm) that will give the largest printed area.

width  = cm

height = cm

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle.

How much wire (in m) should be used for the square in order to maximize the total area?

M = 0

How much wire (in m) should be used for the square in order to minimize the total area?

M= ?

I've tried 16.8 but its wrong.

A firm produces two commodities, ×, and y. The demand functions are

P, = 900 - 2x - 2 y

and

P, = 1400 - 2x -4y

respectively, where P, is the price of commodity x and P, is the price of commodity y. The costs

are given by

C, = 7000 + 100x + x2

and

C, = 10000 + 6 y*

a) Show that the firm's profit function is given by

7 (x, y) = -3x? - 10y? 4xy + 800x +1400y -17000

b) Suppose the firm is required to produce a total of exactly 60 units. Find the values of x

and y that maximize profits.

Use Green’s Theorem to evaluate

∮C(x − 2y2) dx + (y4 + 2xy) dy where C consists of the line segment

from (0, 2) to (0, 4), followed by the curve with parametric equations x = 4 cos t, y = 4 sin t from (0, 4) to (−2, 2√3), then the line segment from (−2, 2√3) to (−1, √3), and finally the curve with parametric equations x = 2 sin t, y = 2 cos t from (−1, √3) to (0, 2).

Find two numbers whose sum is 24 such that the sum of the square of one plus six times the other is a minimum.

A small jewelry box with square of base is to have a volume of 125 cu.cm. Find its dimensions to require the least amount of material.

Determine the second derivative of g(x)=sin(2x³-9x)