Section 4.6 Enhanced Assignment

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 Assignment: Section 4.6 Enhanced Assignment 1. The given figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 30 units long. ​(a) Express the y-coordinate of P in terms of x.​(Hint: Write an equation for the line​AB.) ​(b) Express the area of the rectangle in terms of x. ​(c) What is the largest area the rectangle can​have, and what are its​dimensions? y B P(x,?) A x 0 -15 ​(a) The expression for the​y-coordinate of P is 15 − x x 15 . ​(Type an expression using x as the​variable.) ​(b) The expression for the area of the rectangle is 2x(15 − x) . ​(Type an expression using x as the​variable.) 225 ​(c) The largest area the rectangle can have is 2 square units. ​(Type an integer or a simplified​fraction.) What are the dimensions of the rectangle having the largest​area? A. 15 × B. 15 2 15 2 units × 30 units C. 15 × 30 units 2. A rectangle is constructed with its base on the​x-axis and two of its vertices on the parabola y = 49 − x2 . What are the dimensions of the rectangle with the maximum​area? What is the​area? The shorter dimension of the rectangle is 8.08 and the longer dimension is 32.67 . ​(Round to two decimal places as​needed.) The area of the rectangle is 263.97 . ​(Round to two decimal places as​needed.) 3. You are planning to make an open rectangular box from a 17​-in.-by-33​-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this​way, and what is its​volume? The dimensions of box of maximum volume are 25.86,9.86,3.57 in. ​(Round to the nearest hundredth as needed. Use a comma to separate answers as​needed.) The maximum volume is 3 910.28 in . ​(Round to the nearest hundredth as​needed.) 4. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a​single-strand electric fence. With 700 m of wire at your​disposal, what is the largest area you can​enclose, and what are its​dimensions? The maximum area of the rectangular plot is 61250 The length of the shorter side of the rectangular plot is The length of the longer side of the rectangular plot is 2 m . 175 350 m. m. 5. A covered box is to be made from a rectangular sheet of cardboard measuring 10 inches by 16 inches. This is done by cutting out the shaded regions of the figure and then folding on the dotted lines. What are the dimensions​x, y, and z that maximize the​volume? x= 2 inches y= 6 inches z= 6 inches 6. A right triangle whose hypotenuse is 7 m long is revolved about one of its legs to generate a right circular cone. Find the​radius, height, and volume of the cone of greatest volume that can be made this way. The height of the cone is 1.5 m. 2.2 m. ​(Round to the nearest​tenth.) The radius of the cone is ​(Round to the nearest​tenth.) The volume of the cone is 7.6 3 m . ​(Round to the nearest​tenth.) 7. The height​(feet) of an object moving vertically is given by s = − 16t2 + 160t + 120​, where t is in seconds. Find the​object's velocity at t = 8​, its maximum height and when it​occurs, and its velocity when s = 0. The velocity of the object at t = 8 seconds is − 96 ​(Simplify your answer. Type an integer or a​decimal.) ​ft/second. The maximum height occurs at t = 5 seconds. ​(Simplify your answer. Type an integer or a​decimal.) The maximum height is 520 feet. ​(Simplify your answer. Type an integer or a​decimal.) The velocity when s = 0 is − 182.4 ​(Round to the nearest​hundredth.) ​feet/second. 8. It costs c dollars each to manufacture and distribute backpacks. If the backpacks sell at x dollars​each, the number sold is given by 8a n= + b(97 − x)​, where a and b are positive constants. What selling price will bring a maximum​profit? x−c Express the total profit P in terms of x. 2 ​P(x) = 8a + 97bx − 97bc − bx + bcx The selling price that will maximize profit is ​$ 97 + c 2 . 3 2 9. Suppose that c(x) = 2x − 20x + 14,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. The production level that minimizes the average cost of making x items is x = ​(Simplify your​answer.) At this​level, the average cost of making x items is ​$ ​(Simplify your​answer.) 13950 . 5 . 10. a. How close does the curve y = x come to the point 35 2 ,0 ​? ​(Hint: If the square of the distance is​minimized, square roots can be​avoided.) b. Graph the distance function​D(x) (the square of the​distance) and y = with the result in part​(a). x together and reconcile the result 35 2 a. First determine the function​D(x), the square of the​distance, from the curve y = x to the point 35 2 ,0 ,0 ​, whose value must be minimized. ​D(x) = x− 35 2 2 +x The minimum distance from the curve y = 35 x to the point 2 69 ,0 is units. 2 ​(Type an exact​answer.) b. Graph the distance function​D(x) (the square of the​distance) and y = A. B. y x together. Choose the correct graph below. C. y D. y x y x x x Reconcile the result with part (a). The minimum distance is from the point x= 17 35 2 ,0 to the point 17, 17 on the graph of y = x and this occurs at the value ​, where​D(x), the distance​squared, has its minimum value. ​(Type an ordered pair. Simplify your​answers.) 11. If the price charged for a candy bar is​p(x) cents, then x thousand candy bars will be sold in a certain​city, where ​p(x) = 24 − many candy bars must be sold to maximize​revenue? A. 384 candy bars B. 384 thousand candy bars C. 768 thousand candy bars D. 768 candy bars x 32 . How 12. Find the number of units that must be produced and sold in order to yield the maximum​profit, given the equations for revenue and cost shown below. ​ (x) = 20x − 0.5x R ​C(x) = 6x + 9 2 A. 23 units B. 26 units C. 14 units D. 15 units 13. Find the number of units that must be produced and sold in order to yield the maximum​profit, given the equations for revenue and cost shown below. ​R(x) = 3x 2 ​C(x) = 0.001x + 1.2x + 20 A. 4,200 units B. 2,100 units C. 900 units D. 1,800 units 14. Suppose that ​c(x) = 3x3 − 20x2 + 7,706x is the cost of manufacturing x items. If​possible, find a production level that will minimize the average cost of making x items. A. 7,706 items B. 95 items C. There is not a production level that will minimize average cost. D. 81 items