1. John goes 360km in two hours less than it takes for Kenneth to travel the same distance. If Kenneth travels 30km slower than John, find their speed.a. If x is the number of hours traveled by Kenneth, how do you represent John's traveling time?b. How do you represent Kenneth's speed?c. What equation is formed based on the given problem?d. What is their speed in kilometres per hour?2.Four times a number is five less than the square of the number. What is the number?3.The area of a rectangular garden is 120 sq. m while it's perimeter is 46 m. What are its length and width?a. Let w be the width of the garden. Express the length in terms of the width.b. What is the equation of the areac. What are the dimensions of the garden(w/ solutions )thx!
Answer (1)
Step-by-step explanation:Problem 1:a. If x is the number of hours Kenneth travels, John's traveling time is represented as x - 2 hours.b. Kenneth's speed is represented as 360/x km/hr.c. Let John's speed be J and Kenneth's speed be K. We know that J = K + 30. We also know that John's time is 2 hours less than Kenneth's time. Therefore, we can set up the following equation:360/J = 360/K - 2Substituting J = K + 30, we get:360/(K+30) = 360/K - 2d. To solve for K:Find a common denominator:[360K - 360(K+30)] / [K(K+30)] = -2Simplify:[360K - 360K - 10800] / [K(K+30)] = -2-10800 / [K(K+30)] = -2Solve for K:10800 = 2K(K+30)5400 = K² + 30KK² + 30K - 5400 = 0Factor the quadratic equation:(K - 60)(K + 90) = 0Solve for K:K = 60 or K = -90 (Since speed cannot be negative, we discard K = -90)Therefore, Kenneth's speed (K) is 60 km/hr.Find John's speed:J = K + 30 = 60 + 30 = 90 km/hrAnswer: Kenneth's speed is 60 km/hr and John's speed is 90 km/hr.Problem 2:Let the number be 'n'.Translate the problem into an equation:4n = n² - 5Rearrange the equation into a quadratic equation:n² - 4n - 5 = 0Factor the quadratic equation:(n - 5)(n + 1) = 0Solve for n:n = 5 or n = -1Answer: The number is 5 or -1.Problem 3:a. Let w be the width. If the perimeter is 46m, then 2(length + width) = 46. This simplifies to length + width = 23. Therefore, the length can be expressed as: length = 23 - wb. The equation for the area is:Area = length * width = (23 - w) * w = 120c. To find the dimensions:Expand the area equation:23w - w² = 120Rearrange into a quadratic equation:w² - 23w + 120 = 0Factor the quadratic equation:(w - 8)(w - 15) = 0Solve for w:w = 8 or w = 15Find the length for each width:If w = 8, length = 23 - 8 = 15If w = 15, length = 23 - 15 = 8Answer: The dimensions of the garden are 15m by 8m.
