University:
Augusta University
Course:
MATH 4800 | Secondary Mathematics from an Advanced Perspective
Academic year:

2024
Views:

428

Pages:

2
Author:

Autumn Cox

Question 1 Find the area of the regular polygon with the given apothem a and side length s. decagon, a 26 m, s = 16.9 m Answer Use the formula for the area of a regular polygon with n sides with length s and apothem a: A = (1/2)ans Substitute n=10, a=26 m, and s=16.9 m: A = (1/2)(26)(10)(16.9) A = 2197 So, the area is 2197 m². Question 2 Rectangle ABCD has dimensions 1 by 2. The next rectangle PQRS has dimensions 1/2 by 1. In like manner each rectangle has dimensions half of the size of the preceding rectangle. If this sequence of rectangles continues endlessly, what is the sum of the areas of ail the rectangles? Answer The areas of the rectangles are in geometric sequence: 1x2, (1/2)x1, (1/4)x(1/2),... = 2, 1/2, 1/8,... where a₁=2 (first term) and r=1/4 (common ratio). Using the sum of infinite geometric sequence, the sum of the areas is: S = a₁ / (1 - r) S = 2 / (1 - (1/4)) S = 2 / (3/4) = 2 * (4/3) = 8/3 Question 3 The base of a rectangular prism has an area of 19.4 square meters and the prism has a volume of 306.52 cubic meters. Write an equation that can be used to find the height h of the prism. Then find the height of the prism. Answer The volume of a rectangular prism is the product of its base area B and its height h: V = Bh Substituting given, we have the equation: 306.52 = 19.4h Divide both sides by 19.4: h = 306.52 / 19.4 h ≈ 15.8 So, the height of the prism is 15.8 meters. Question 4 Six congruent rectangular prisms volume can be modeled by V(x) below. V(x) = 54x³ +264x² - 738x + 420 Answer Since there are 6 rectangular prisms, we can factor out 6 from the polynomial: 6(9x³ + 44x² - 123x + 70) So, the volume of one rectangular prism is: 9x³ + 44x² - 123x + 70 Question 5 A packing company is doing an inventory of boxes. Their most popular box is display below. You can use the formula V = lwh to find the volume of the box. The volume of the box is 40 ft³. What is the value of x? Find the length and the width of the box. Describe any extraneous solutions. Answer From the figure, l = 3x – 5 ft, w = 2x - 1 ft, and h = 2 ft. The volume is 40 ft³ so we can write: V = lwh 40 = (3x - 5)(2x - 1)(2) Divide both sides by 2: 20 = (3x - 5)(2x - 1) Expand the right side: 20 = 6x² - 13x + 5 Write in standard form: 0 = 6x² - 13x - 15 Factor the right side: 0 = (6x + 5)(x - 3) By zero-product property, x = -5/6, 3 Since x = -5/6 gives negative length and width, then it is extraneous. Hence, x = 3 Using the value of x, the length is 3(3) - 5 = 4 ft and the width is 2(3) - 1 = 5 ft.