Modeling #6 - Questions and Answers

Question 1 The area (in square centimeters) of a square coaster can be represented by d² + 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) a = d² + 8d + 16 = d² + 2(d)(4) + 4² (a+b)² = a² + 2ab + b² d² + 2(d)(4) + 4² = (d + 4)² Area = (side)² (side)² = (d + 4)² side = (d + 4) Answer: the length of the playground is (d+4) b) In part a we find the side of the square. Perimeter of square is 4 × side P = 4(d + 4) P = 4d + 16 Answer is 4d + 16. Question 2 A cheetah that is running 90 feet per second is 120 feet behind an antelope that is running 60 feet per second. How long will it take cheetah to catch up to the antelope? Answer 1 Formula Used: Distance = speed × time Let the time taken by the cheetah to catch up to the antelope be t seconds. The distance traveled by the cheetah in t seconds = 90t The distance traveled by the antelope in t seconds = 60t As, in t seconds the cheetah catches up to the antelope, so, 90t - 60t = 120 30t = 120 t = 120 / 30 t=4 Conclusion: Thus, the time taken by the cheetah to catch up to the antelope is 4 seconds. Answer 2 Given: t = d/v where: t is the time taken, d is the distance between the cheetah and the antelope, and -v is the relative velocity of the cheetah with respect to the antelope. Given that the cheetah is 120 feet behind the antelope, we have d = 120 feet. The relative velocity of the cheetah with respect to the antelope is the difference between their speeds, so v = 90 - 60 = 30 feet per second. Plugging these values into the equation, we get: t = 120 / 30 = 4 seconds. Therefore, it will take the cheetah 4 seconds to catch up to the antelope. Answer 3 Step 1: The distance traveled by the cheetah in t seconds can be calculated by multiplying its speed (in feet per second) by the time: Distance traveled by the cheetah = Speed of the cheetah × t Similarly, the distance traveled by the antelope in t seconds is given by: Distance traveled by the antelope = Speed of the antelope × t Step 2: Given that the cheetah is initially 120 feet behind the antelope, we can set up the following equation: Distance traveled by the cheetah = Distance traveled by the antelope + Initial distance between them 90t = 60t + 120 Simplifying the equation, we get: 30t = 120 Step 3: To solve for t, we divide both sides of the equation by 30: t = 120 / 30 Simplifying further: t = 4 seconds Therefore, it will take the cheetah 4 seconds to catch up to the antelope. Question 3 Write the solve an inequality that represents how many $20 bills you can withdraw from the account without going below the minimum balance. Given information: Current balance is $320 and minimum balance in the account should be $100. Answer 1 Calculation: Let x denote the number of bills that can be withdrawn from the account without going below the minimum balance. According to question, 320 - 20x ≥ 100 ⇒ 20x ≤ 320 - 100 ⇒ 20x ≤ 220 ⇒ x ≤ 220/20 ⇒ x ≤ 11 So, x ≤ 11 is the inequality that denotes how many $20 bills can be withdrawn from the account without going below the minimum balance. Answer 2 Given information: Current balance = 320 Minimum balance = 100 When you withdraw x 20 dollar bills, the amount you will withdraw is 20x. To ensure that the resulting balance does not go below the minimum balance, we can set up the following inequality: 320 - 20x ≥ 100 Let's solve this inequality to find the range of values for x: 320 - 20x ≥ 100 -20x ≥ 100 - 320 -20x ≥ -220 x ≤ -220 / -20 x ≤ 11 Therefore, the inequality that represents how many 20 dollar bills you can withdraw from the account without going below the minimum balance is x ≤ 11. Answer 3 Solution: Let's assume the number of 20 dollar bills to be withdrawn is represented by the variable x. The total amount to be withdrawn can be expressed as 20x. The current balance in the account is 320. To ensure that the balance does not go below the minimum balance of 100, we can set up the following inequality: 320 - 20x ≥ 100 Now, let's solve this inequality: Step 1: Subtract 320 from both sides of the inequality: -20x ≥ 100 - 320 Step 2: Simplify the right side of the inequality: -20x ≥ -220 Step 3: Divide both sides of the inequality by -20 (note that we flip the direction of the inequality because we are dividing by a negative number): x ≤ -220 / -20 Simplifying further: x ≤ 11 Question 4 You trimmed a large square picture so that you could fit it into a frame. The area of the cut picture is 20 square inches. What is the area of the original picture? Answer Calculation: Area is: (x - 5)(x - 6) = 20 x² - 5x - 6x + 30 - 20 = 0 x² - 11x + 10 = 0 x² - 10x - x + 10 = 0 x(x - 10) - 1(x - 10) = 0 (x - 10)(x - 1) = 0 x - 10 = 0 or x - 1 = 0 x = 10 or x = 1 x = 10 Area of larger square is = 10 * 10 = 100. Question 5 The normal body temperature of a camel is 37°C. This temperature varies by up to 3° throughout the day. Write and solve an absolute value inequality that represents the range of normal body temperatures (in degrees Celsius) of a camel throughout the day. Given information: Normal temperature of camel = 37°C This temperature can vary by up to 3°C Answer 1 Let T denote the temperature (in degrees Celsius) of a camel throughout the day. Now it is given that the normal temperature of the camel is 37°C and it can vary by up to 3°C. Thus, the required absolute value inequality is: |T - 37| ≤ 3 Now consider, |T - 37| ≤ 3 -3 ≤ T - 37 ≤ 3 -3 + 37 ≤ T - 37 + 37 ≤ 3 + 37 34 ≤ T ≤ 40 The solution is 34 ≤ T ≤ 40. Answer 2 We can write the absolute value inequality: |x - 37| ≤ 3 Here, x represents the body temperature of the camel throughout the day. The absolute value of the difference between x and 37 should be less than or equal to 3, indicating that the temperature can vary within 3 degrees Celsius from the normal temperature. To solve this absolute value inequality, we can break it down into two separate inequalities: x - 37 ≤ 3 -(x - 37) ≤ 3 Solving the first inequality: x - 37 ≤ 3 Adding 37 to both sides: x ≤ 3 + 37 Simplifying: x ≤ 40 Solving the second inequality: -(x - 37) ≤ 3 Expanding the negative sign: -x + 37 ≤ 3 Subtracting 37 from both sides: -x ≤ 3 - 37 Simplifying: -x ≤ -34 Multiplying both sides by -1 (which reverses the inequality): x ≥ 34 Therefore, the solution to the absolute value inequality |x - 37| ≤ 3 is given by 34 ≤ x ≤ 40. Answer 3 Explanation: We have: |x - 37| ≤ 3 In this inequality, x represents the body temperature of the camel throughout the day. The absolute value of the difference between x and 37 must be less than or equal to 3. To solve this absolute value inequality, we can consider two cases: Case 1: x - 37 ≥ 0 In this case, the absolute value can be removed, and we have: x - 37 ≤ 3 Solving for x: x ≤ 3 + 37 x ≤ 40 Case 2: x - 37 < 0 In this case, we need to negate the absolute value by multiplying it by -1, and we have: -(x - 37) ≤ 3 Simplifying the inequality: -x + 37 ≤ 3 Solving for x: -x ≤ 3 - 37 -x ≤ -34 Remember that when multiplying or dividing an inequality by a negative number, the inequality sign must be flipped. So, multiplying both sides by -1: x ≥ 34 Therefore, the solution to the absolute value inequality is: x ∈ [34, 40] This means that the normal body temperature of a camel throughout the day can range from 34 degrees Celsius to 40 degrees Celsius, inclusive. Answer 4 The absolute value of the difference between x and the normal temperature of a camel, 37 degrees Celsius, should be less than or equal to 3 degrees Celsius. Mathematically, we can express this as: |x - 37| ≤ 3 This inequality states that the absolute value of the difference between x and 37 is less than or equal to 3. To solve this inequality, we can break it down into two separate inequalities: x - 37 ≤ 3. This represents the case when x - 37 is positive. -(x - 37) ≤ 3. This represents the case when x - 37 is negative. Solving the first inequality: x - 37 ≤ 3 x ≤ 3 + 37 x ≤ 40 Solving the second inequality: -(x - 37) ≤ 3 -x + 37 ≤ 3 -x ≤ 3 - 37 -x ≤ -34 34 ≤ x Combining the solutions, we find that the range of normal body temperatures of a camel throughout the day is 34 degrees Celsius (or greater) to 40 degrees Celsius (or less). Therefore, the absolute value inequality that represents the range of normal body temperatures of a camel throughout the day is: 34 ≤ x ≤ 40