Math Unit 3

Consider the net of a triangular prism where each unit on the coordinate plane represents four feet. If a sheet of plywood measures 4 ft x 8 ft, how many sheets of plywood will a carpenter need to build the prism? A) 3 B) 3.5 C) 4 D) 4.5

3.5 each unit = 4 feet each triangle 1 2 (4 x 4) = 8 ft2 each rectangle 4 x 8 = 32 ft2 then, 2 x 8 = 16 ft2 3 x 32 = 96 ft2 16 + 96 = 112 ft2 and a sheet of plywood measures: 4 x 8 = 32 ft2 thus, 112 ÷ 32 = 3.5 sheets of plywood

A farmer wants to build a pen for his sheep. One side of the pen will be a river. The sheep need 2000 m2 of area to graze. The farmer wants to use the least amount of fencing as possible. Which equation should the farmer use to find the minimum amount of fencing? A) F = 4000 x + x B) F = 2000 x + x C) F = 4000 + x D) F = 2000 + x/x

A)The area of the pen is yx = 2000, y = 2000 x . The farmer needs 2y + x of fencing. Substitute in the value of y. F = 2( 2000 x ) + x or F = 4000 x + x

Po has a frame that is 9 in x 8 in. What is the maximum area the picture can have to fit in the frame? A) 63 sq in B) 72 sq in C) 81 sq in D) 96 sq in

The correct answer is found by multiplying the length by the width. 9 × 8 = 72 sq in.

A stained glass window has been designed in the pattern shown using regular polygons. Find the measure of the marked angle. A) 60° B) 90° C) 150° D) 170°

The solution is 150°. To get the entire angle, add the angle in the square 90° to the angle in the triangle 60°.

Rosario draws the right triangular prism shown here and calculates the volume. She then draws a second right triangular prism in which the dimensions are doubled. What is the relationship between the volumes of the two prisms? A) The volume of the second prism is 2 times that of the first prism. B) The volume of the second prism is 4 times that of the first prism. C) The volume of the second prism is 6 times that of the first prism. D) The volume of the second prism is 8 times that of the first prism.

The volume of the second prism is 8 times that of the first prism. We can calculate the volumes of the prisms using V = Bh, where B is the area of the base. V(1st prism) = 5(7) = 35 in3 V(2nd prism) = 20(14) = 280 in3

A plane is to be loaded with bottles of water and medical supplies to be sent to victims of an earthquake. Each bottle of water serves 10 people and each medical kit aids six people. The goal is to maximize z, the total number of people helped, where z = 10x + 6y, and x is the number of bottles and y is the number of medical kits. Using the constraints of the situation (plane weight and volume capacities) the shaded region is in the graph is obtained. Which vertex of the region represents the solution to the maximization problem? A) (0, 6000) B) (4000, 0) C) (2000, 4000) D) (6000, 8000)

(2000, 4000) is correct. Test each of the 4 vertices [(0, 0), (0, 6000), (4000, 0), and (2000, 4000)] in the z function to get the largest answer.

An open box will be made from a rectangular piece of cardboard that is 8 in. by 10 in. The box will be cut on the dashed red lines, removing the corners, and then folded up on the dotted lines. The box needs to have the MAXIMUM volume possible. How long should the cuts be? A) 1.5 in. B) 5.8 in. C) 52 in. D) 80 in.

1.5 -The height, width, and length of the box will be (x)(8 - 2x)(10 - 2x). The volume of the box will be 4x3 - 36x2 + 80x = 0 Use a graphing calculator to graph the polynomial. Use the maximum feature to see the greatest volume the box can have. Be sure to set your window accordingly and think about what constraints are on the box. The maximum volume for the box is 52 in3. The cut would be 1.5 in. long. This is where the graph crosses the x-axis.

The profit of a company in thousands of dollars is given by the quadratic function P(x) = 3000 + 1200x - 6x2 where x is the amount, in thousands, the company spends on advertising. Find the amount, x, that the company has to spend to maximize its profit. A) 100 B) 1,000 C) 10,000 D) 100,000

100 is correct. Function P that gives the profit is a quadratic function with the leading coefficient a = -6. This function (profit) has a maximum value at x = h = -b/2a; thus, x = h = -1200/2(-6) = 100

A steel plant has two sources of ore, source A and source B. In order to keep the plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to no more than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 300 lbs of steel per ton, and ore from source B yields 400 lbs of steel per ton, how many tons of ore from each source should be processed each day to maximize the amount of steel produced? A) 1 ton from source A, 2 tons from source B B) 2 tons from source A, 4 tons from source B C) 3 tons from source A, 0 tons from source B D) 4 tons from source A, 0 tons from source B

2 tons from source A, 4 tons from source B is correct. The three inequalities are a+b ≤ 3, b ≤ 2a, and 20a + 10b ≤ 80. They form a quadrilateral (with the x-axis) having vertices (1,2), (2,4), (3,0), (4,0). Test each of these in the production function, P = 300a + 400b to get the answer.

A farmer needs to build a pen for his sheep. One side of the pen will use a fence already standing. The sheep need 2000 square meters of area to graze. What width y should the farmer build to use the LEAST amount of fencing for his sheep pen? (to the nearest tenth) A) 27.4 m B) 29.5 m C) 31.6 m D) 33.5 m

31.6 m The area of the pen: yx = 2000 y = 2000/x The farmer needs 2y + x fencing: Substitute in the value of y = 2000/x F = 2(2000/x) + x F = 4000/x+ x Graph this function on a graphing calculator. The minimum value occurs at x = 63.246 Plug 63.246 back into the original equation xy = 2000 y = 2000/x y = 2000/63.246 = 31.622 Therefore, a width of 31.6 m would use the least amount of fencing.

Trina has $1000 to purchase an open-top cylindrical dog pen in her backyard. She wants the height of the pen to be 5 feet. If the pen costs $1 per square foot, what is the biggest pen (in terms of the radius) that she can afford? Round your answer to the nearest hundredth of a foot. A) 7.96 feet B) 15.92 feet C) 31.83 feet D) 63.66 feet

31.83 feet is correct. The surface area of the pen is given by SA = 2πrh Since she can afford $1000 and the cost is $1/square foot, then the surface area should be equal to $1000 to find the largest pen. 1000 = 2π(r)(5) 100 = πr r ≈ 31.83 ft

A tent is in the shape of a triangular prism. The front and back are isosceles triangles with base 6 feet and height 4 feet. The surface area of the entire tent is 104 square feet. What is the depth of the tent? A) 5 feet B) 8 feet C) 16 feet D) 18.4 feet

5 feet is correct. The surface area is 12 + 12 + 5x + 5x + 6x = 16x + 24 = 104, so x = 5.

Consider the net of a triangular prism where each unit on the coordinate plane represents five feet. If a can of spray paint covers 25 square feet, how many cans of spray paint are needed to paint the outside of the prism blue? A) 5 cans B) 7 cans C) 10 cans D) 14 cans

7 each unit = 5 feet each triangle 1 2 (5 x 5) = 12.5 ft2 each rectangle 5 x 10 = 50 ft2 then, 2 x 12.5 = 25 ft2 3 x 50 = 150 ft2 25 + 150 = 175 ft2 thus, 175 ÷ 25 = 7 spray cans

Consider the net of a triangular prism where each unit on the coordinate plane represents ten feet. If a can of spray paint covers 50 square feet, how many cans of spray paint are needed to paint the outside of the prism red? A) 8 cans B) 10 cans C) 12 cans D) 15 cans

8

Organizers of an outdoor concert will use 320 feet of fencing to fence off a rectangular VIP section. What is the maximum area that they can fence off? A) 3,200 square feet B) 6,400 square feet C) 12,800 square feet D) 24,000 square feet

Since the perimeter of the VIP section will be 320 feet, the sum of the length and the width will be feet. For this reason, if the length is l feet, the width will be 160 - l feet. The area of the VIP section in square feet can be modeled by the function A = l(160 - l) = 160l - l2, which can be rewritten as A = -(l2 - 160l), and by completing the square, the function becomes A = -(l - 80)2 + 6,400. The maximum area that the organizers can fence off is the maximum of the function, which is 6,400 square feet.