Modeling #5 - Questions and Answers

Question 1 The area of the patio is 216 square feet. a. Write and equation that represents the area of the patio. b. Find the dimensions of the patio by completing the square. Answer a) Area of the patio is 216 ft² Length (l) of the patio is (x + 6) ft Width (b) of the patio is x ft We use the area of a rectangle = lb Area of the patio = 216 Length (l) of the patio = (x + 6) Width (b) of the patio = x 216 = (x + 6)(x) x² + 6x = 216 x² + 6x - 216 = 0 b) Area of a rectangle = lb Area of the patio = 216 Length (l) of the patio = (x + 6) Width (b) of the patio = x 216 = (x+6)(x) x² + 6x = 216 x² + 6x - 216 = 0 x² + 18x - 12x - 216 = 0 x(x + 18) - 12(x + 18) = 0 (x + 18)(x - 12) = 0 (x + 18) = 0 (x - 12) = 0 x = -18 x = 12 width cannot be negative So Width (b) of patio = 12 ft Length (l) of patio = 18 ft Question 2 The function y = -16x² + 36 y (in feet) is the height of an apple x seconds after falling from a tree. Determine the x- and y- intercepts. Answer Given: The height in y (in feet) of an apple x seconds after falling from a tree given by the function: f(x) = -16x² + 36 .......(1) Calculation: The x-intercept of the given function is found by setting y to 0 and solving for f (x) = 0 in equation (1), f(x) = -16x² + 36 0 = -16x² + 36 16x² = 36 x² = 36 / 16 = 9 / 4 x = ± √9/4 = ± 3/2 x = ± 1.5 Now, here x stands for time and time cannot be negative, so the solution is x = 1.5. Thus, the x-intercept of given function is (1.5, 0). The x-intercept in this case provides the time in which apple will hit the ground. The y-intercept of the given function is found by substituting x = 0 in equation (1), f(x) = -16x² + 36 y = -16(0)² + 36 y = 0 + 36 y = 36 Thus, the y-intercept of the given function is (0, 36) The y-intercept in this case gives the initial height of the apple. Question 3 The football field is rectangular. a. Write a polynomial that represents the area of the football field. b. Find the area of the football field when the width is 160 feet Given information: a) A rectangular field whose length and breadth are (10x+10)ft and (4x+20) ft. b) If width = 160 feet. Answer Calculations: a) Area of the football field: = Length * Breadth = (10x + 10) (4x +20) = 40x² + 200x + 40x + 200 = 40x² + 240x + 200 b) width = 160 feet (4x + 20) = 160 4x = 140 x = 35 area = 40x² + 200x + 40x + 200 = 40(35²) + 200(35) + 40(35) + 200 = 49000 + 7000 + 1400 + 200 = 57600 Question 4 The edge length s of a cube with a surface area of A is given by s = √(A/6). What is the surface area of a cube with an edge length of 4 inches? Answer Given information: s = √(A/6) and s = 4 Formula Used: Simplify. Calculation: s = √(A/6) As s = 4 So, 4 = √(A/6) Now, squaring both sides (√(A/6))² = (4)² So, simplify A/6 = 16 Now, A = 16 × 6 Hence, A = 96. Question 5 Write and solve a compound inequality that represents the possible temperatures (in degrees Fahrenheit) of the interior of the iceberg. Given: The temperatures are given -20 and -15 in degree centigrade. Answer Formula used: C = 5/9 (F – 32) Calculation: For C = -20° -20 = 5/9 (F – 32) F = 9/5 (-20) + 32 = -36 + 32 F = -4° For C = -15° F = 9/5 (-15) + 32 F = -27 + 32 F = 5° The value of temperature at C = -20° is F = -4° and at C = -15° is F = 5° Hence, the compound inequality of the temperature is found -4° ≤ F ≤ 5° Answer 2 To represent the possible temperatures of the interior of the iceberg in degrees Fahrenheit, we need to convert the given temperatures from Celsius to Fahrenheit. The formula for converting Celsius to Fahrenheit is: F = (9/5)C+32 where F represents the temperature in Fahrenheit and C represents the temperature in Celsius. Given that the temperatures are -20°C and -15°C, we can calculate the corresponding temperatures in Fahrenheit: For -20°C: F = (9/5)(-20) + 32 = -4 For -15°C: F = (9/5)(-15) + 32 = 5 Therefore, the possible temperatures of the interior of the iceberg in degrees Fahrenheit can be represented by the compound inequality: -4 ≤ T ≤ 5 where T represents the temperature in degrees Fahrenheit. Answer 3 First, let's convert the given temperatures to degrees Fahrenheit. The conversion formula from Celsius to Fahrenheit is: F = (9/5)C + 32. For -20°C, the temperature in Fahrenheit is: F = (9/5)(-20) + 32. Simplifying this equation gives: F = -36 + 32 = -4°F. For -15°C, the temperature in Fahrenheit is: F = (9/5)(-15) + 32. Simplifying this equation gives: F= -27 + 32 = 5°F. Now, let's write and solve a compound inequality to represent the possible temperatures: -4 ≤ F ≤ 5 Therefore, the compound inequality that represents the possible temperatures (in degrees Fahrenheit) of the interior of the iceberg is: -4 ≤ F ≤ 5. Answer 4 Explanation: Given that the temperatures are -20°C and -15°C, we can substitute these values into the conversion formula to find their Fahrenheit equivalents: F₁ = (9/5)(-20) + 32 F₂ = (9/5)(-15) + 32 Simplifying these equations will give us the temperatures in degrees Fahrenheit: F₁ = -4 F₂ = 5 Now we can write a compound inequality to represent the possible temperatures within this range. Let's assume T represents the temperature in degrees Fahrenheit. The compound inequality can be written as: -4 ≤ T ≤ 5 This inequality indicates that the temperature T is greater than or equal to -4°F and less than or equal to 5°F. Thus, any temperature within this range is a possible temperature for the interior of the iceberg.