Calculus

Question

Please help me with these five problems. Do what you can, answer and work will be greatly appreciated. Thanks a lot! 1.Suppose we want to construct a box with a top. We want the box to have a capacity of 300 cubic units. For economic reasons, we would like to use the least amount of material to construct the box. Let the dimensions of the base be x by y and the height be h. Draw the box. Write the equation of the surface area of the box as a function of the two variables x and y. 2. A cylindrical tank 4 ft in diameter fills with water at the rate of 10 ft/s. express the depth of the water in the tank as a function of the time t in seconds. Assume the tank is empty at time t=0. 3.A storeowner bought x dozen toy dolls at a cost of $4.00 per dozen, and sold them at $.85 a piece. Express the profit P (in dollars) as a function of x. 4.Two sides of a rectangle lie along the axes as shown at the right. One vertex is a point on the line 3x+4y=24. a. Express the area A of the rectangle as a function of the x-coordinate of P. b. What is the domain of the area function? c. What is the maximum area of A? 5. Triangle OAB is an isosceles triangle with vertex O at the origin and vertices A and B on the part of the parabola y=8-x^2 that is above the x-axis. a. Express the area of the triangle as a function of the x-coordinate of A. b. What is the domain of the area function? c. Use a calculator to find the maximum area. Thankyou.

Answer

Question #1 The equation for the box's area is A=2xh+2yh+2xy The volume = 300 = xyh, so h=300/(xy) Substitute expression for h into equation for A: A=600/y+600/x+2xy Question #2 Depth = 10t where Depth is in ft and t is in seconds Question #3 The storeowner paid $4.00 per dozen and received 12($0.85) = $10.20 per dozen for a profit of $6.20 per dozen. P=x($6.20) Question #4 The y value of the Point P is found by solving the equation 3x+4y=24 for y: y=6-3x/4 The equation for the area of the rectangle is A=xy Substituting the expression for y into the equation for A: A=6x-(3x^2)/4 A is zero for x=0 and x=8, so the domain is 0