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Answer (1)
Answer:1. The length of a rectangle is 6 inches more than its width. The area of the rectangle is 55 in², find the dimensions of the rectangle. Let: - w represent the width of the rectangle- l represent the length of the rectangle We know: - l = w + 6 (The length is 6 inches more than the width)- l * w* = 55 (The area of a rectangle is length times width) Now we can substitute the first equation into the second equation: - (w + 6) * w = 55 Expand the equation: - w² + 6w = 55 Move all terms to one side: - w² + 6w - 55 = 0 Factor the quadratic equation: - (w + 11) (w - 5) = 0 Solve for w: - w + 11 = 0 or w - 5 = 0- w = -11 or w = 5 Since the width cannot be negative, we discard w = -11. Therefore, the width of the rectangle is w = 5 inches. Now, we can find the length: - l = w + 6- l = 5 + 6- l = 11 inches Therefore, the dimensions of the rectangle are 5 inches by 11 inches. 2. The sum of two numbers is 14, and the product of these two numbers is 45. What are the numbers? Let: - x represent the first number- y represent the second number We know: - x + y = 14- x * y = 45 We can solve this system of equations using substitution: 1. Solve the first equation for x: - x = 14 - y2. Substitute this value of x into the second equation: - (14 - y) * y = 453. Expand the equation: - 14y - y² = 454. Move all terms to one side: - y² - 14y + 45 = 05. Factor the quadratic equation: - (y - 9) (y - 5) = 06. Solve for y: - y - 9 = 0 or y - 5 = 0- y = 9 or y = 57. Substitute each value of y back into the equation x = 14 - y to find the corresponding values of x: - If y = 9, then x = 14 - 9 = 5- If y = 5, then x = 14 - 5 = 9 Therefore, the two numbers are 5 and 9. 3. The length of a rectangular garden is 4 meters more than the width. If the area is 96 m², find the dimensions of the rectangular garden. Let: - w represent the width of the garden- l represent the length of the garden We know: - l = w + 4 (The length is 4 meters more than the width)- l * w = 96 (The area of a rectangle is length times width) Now we can substitute the first equation into the second equation: - (w + 4) * w = 96 Expand the equation: - w² + 4w = 96 Move all terms to one side: - w² + 4w - 96 = 0 Factor the quadratic equation: - (w + 12) (w - 8) = 0 Solve for w: - w + 12 = 0 or w - 8 = 0- w = -12 or w = 8 Since the width cannot be negative, we discard w = -12. Therefore, the width of the garden is w = 8 meters. Now, we can find the length: - l = w + 4- l = 8 + 4- l = 12 meters Therefore, the dimensions of the rectangular garden are 8 meters by 12 meters.
