Questions and Answers #4 Modeling

Questions and Answers Sheet 4 Modeling Question #1 You and your friend drive toward each other. The equation 50h=190-45h represents the number h of hours until you and your friend meet. When will you meet? Answer: Calculation: In order to find the number of hours you and your friend meet, we need to isolate given equation 50h=190-45h for h. 50h=190-45h Adding 45h on both sides 50h+45h=190-45h+45h Simplify 95h=190 Dividing both sides by 95 95ℎ 95 = 190 95 ⇒ ℎ = 2. Therefore, you and your friend will meet after 2 hours. Question #2 The function 𝑦 = −16𝑥 2 + 36 represents the height y (in feet) of an apple x seconds after falling from a tree. Find and interpret 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) = −16𝑥 2 + 36 … … . . (1) Calculation: The x-intercept of the given function is found by substituting f(x)=0 in equation (1), 𝑓(𝑥) = −16𝑥 2 + 36 0 = −16𝑥 2 + 36 16𝑥 2 = 36 36 𝑥2 = 𝑥2 = 16 9 4 𝑥=± 3 2 𝑥 = ±1.5 Now, here x represents time and time can never be negative, so solution is x=1.5 . Thus, the x-intercept of given function is (1.5, 0). The x-intercept in this case gives 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) = −16𝑥 2 + 36 𝑦 = −16(0)2 + 36 𝑦 = 0 + 36 𝑦 = 36 Thus, the y-intercept of 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) = 40𝑥 2 + 200𝑥 + 40𝑥 + 200 = 40𝑥 2 + 240 + 200 b) width =160 feet (4𝑥 + 20) = 160 4𝑥 = 140 𝑥 = 35 area = 40𝑥 2 + 200𝑥 + 40𝑥 + 200 = 40(30 ⋅ 30) + 200(35) + 40(35) + 200 = 4900 + 8600 = 57600 Question #4 𝐴 The edge length s of a cube with a surface area of A is given by 𝑠 = √ . What is the surface area of a 6 cube with an edge length of 4 inches? Answer: 𝐴 GIven information: 𝑠 = √ and s=4 6 Formula Used: Simplify. Calculation: 𝑠 = √ 𝐴 6 As s=4 So, 4 = √ 𝐴 6 Now, squaring both sides 2 𝐴 (√ ) = (4)2 6 So, simplify 𝐴 = 16 6 Now, 𝐴 = 16 × 6 Hence, A=96 . Question #5 The expression 4 3 𝜋𝑟 3 represents the volume of a sphere with radius r. Why is this expression a monomial? What is its degree? Answer: Calculation: The volume is 4 3 4 3 𝜋𝑟 3 𝜋𝑟 3 is only 1 term. Therefore it is monomial. The maximum exponent of 4 3 𝜋𝑟 3 is 3 Therefore degree is 3. Question #6 The area (in square centimeters) of a square coaster can be represented by 𝑑 2 + 8𝑑 + 16. a. Write and expression that represents the side length of the coaster. b. Write and expression for the perimeter of the coaster. Answer: a) Calculation: 𝑎 = 𝑑 2 + 8𝑑 + 16 = 𝑎 = 𝑑 2 + 8𝑑 + 16 = 𝑑 2 + 2(𝑑)(4) + 42 𝑎2 + 2𝑎𝑏 + 𝑏 2 = (𝑎 + 𝑏)2 𝑑 2 + 2(𝑑)(4) + 42 = (𝑑 + 4)2 Area = (𝑠𝑖𝑑𝑒)2 Compare the area (𝑠𝑖𝑑𝑒)2 = (𝑑 + 4)2 side =(d+4) Thus the length of playground is (d+4) b) Calculation: In part a we find side of square. Perimeter of square is 4 × side Perimeter =4(d+4) Perimeter =4d+16 Thus the perimeter is 4d+16 . Question #7 A stained glass artist has a fixed cost of $150. It costs the artist $15 to procedure each piece, but each piece sells for $35. The equation C=150+15n represents the total cost, C, for producing n pieces. The total revenue for n pieces is determined by the equation R=35n . What constant is necessary to include when modeling this situation? - n must be greater than or equal to 0, since the number of items created cannot be negative. - R must be greater than or equal to 35, since the number of items will always be positive. - R-C must be less than or equal to 0, since the artist wants to make money. - C must be 150, since there is a fixed cost to produce each piece. Answer: Step 1 Given, Total cost =C=150+15n total revenue = R=35n to make profit we have to increse production R-C>0 35n-15n-150>0 15n>150 n>10 if n>10 or atleast 10 piece have to produce to cross fixed amount. Step 2 R-C must be greater than zero and always positive to make money. n should be always positive. Question #8 The circumference C of the art room in a mansion is approximated by the formula 𝐶 ≈ 2𝜋√ 𝑎2 +𝑏 2 2 . Approximate the circumference of the room. a is 20ft b is 16ft Answer: Calculation: 𝐶 ≈ 2𝜋√ 𝑎2 +𝑏 2 2 a=20 b=16 Substituting value of a and b 𝐶 ≈ 2𝜋√ 𝐶 ≈ 2𝜋√ 𝐶 ≈ 2𝜋√ 202 +162 2 400+256 2 656 2 𝐶 ≈ 2𝜋√328 𝐶 ≈ 2 𝜋 × 18.11 Put value of pi as 3.14 𝐶 ≈ 2 × 3.14 × 18.11 𝐶 ≈ 113.7308 Hence, approximate answer is C=114ft Question #9 Write a recursive rule for the length of the deer antler over time. 1 day 2 days 3 days 4 days 1 3 1 4 in. 4 in. 5in 5 in 2 4 4 Answer: Given: The length of the deer antler grows as n 1 an 1 9 4 = 2 2 2 3 4 3 19 1 21 4 = 5 5 = 4 4 4 4 where n is in days. 9 9 The value of 𝑎𝑛 for n=1 is . Hence we must have 𝑎1 = 2 2 It can be observed that the given values of 𝑎𝑛 forms an arithmetic progression with a common difference 1 of . 4 A recursive relation for an arithmetic progression with common difference d is given by 𝑎𝑛 = 𝑎𝑛−1 + 𝑑. Hence, for the given problem the recursive relation must be given by: 1 9 𝑎𝑛 = 𝑎𝑛−1 + , 𝑎1 = 4 2 Question #10 The function d(t)=300,000t represents the distance (in kilometers) that light travels in t seconds. a. How far does light travel in 15 seconds? b. How long does it take light to travel 12 million kilometers? Answer: a) Given information: d(t)=300000t represents the distance that light travels in t seconds. Here we will put t=1 5 into the given function. d(t)=300000t d(15)=300000(15) d(15)=4500000km