The length of a standard jewel case is 4 cm more than its width. The area of the rectangular top of the case is 320 cm. Find the length and width of the jewel case.
Answer (2)
Let w be the width and l be the length, where l = w + 4 .
Express the area as w ( w + 4 ) = 320 , which simplifies to w 2 + 4 w − 320 = 0 .
Solve the quadratic equation using the quadratic formula to find w = 16 (since width must be positive).
Calculate the length: l = w + 4 = 16 + 4 = 20 . The length of the jewel case is 20 cm and the width is 16 cm. l e n g t h = 20 c m , w i d t h = 16 c m
Explanation
Understanding the Problem Let's analyze the problem. We are given that the length of a jewel case is 4 cm more than its width, and the area of the rectangular top of the case is 320 cm 2 . We need to find the length and width of the jewel case.
Defining Variables and Area Equation Let w be the width of the jewel case and l be the length of the jewel case. We can express the length in terms of the width as l = w + 4 . The area of the rectangle is given by A = l × w = 320 .
Forming a Quadratic Equation Substitute l = w + 4 into the area equation: ( w + 4 ) w = 320 . Expanding this, we get w 2 + 4 w = 320 . Rearranging the equation into a quadratic equation, we have w 2 + 4 w − 320 = 0 .
Solving for Width Now, we solve the quadratic equation w 2 + 4 w − 320 = 0 for w using the quadratic formula: w = 2 a − b ± b 2 − 4 a c , where a = 1 , b = 4 , and c = − 320 . Plugging in these values, we get:
w = 2 ( 1 ) − 4 ± 4 2 − 4 ( 1 ) ( − 320 ) = 2 − 4 ± 16 + 1280 = 2 − 4 ± 1296 = 2 − 4 ± 36
This gives us two possible values for w : w = 2 − 4 + 36 = 2 32 = 16 and w = 2 − 4 − 36 = 2 − 40 = − 20 . Since the width must be positive, we choose the positive solution, so w = 16 cm.
Calculating Length Now we calculate the length l using l = w + 4 . Since w = 16 , we have l = 16 + 4 = 20 cm.
Final Answer and Verification Therefore, the length of the jewel case is 20 cm and the width is 16 cm. Let's check if these values are reasonable: The area is 20 × 16 = 320 cm 2 , which matches the given area. Also, the length is 4 cm more than the width, as stated in the problem.
Examples
Understanding how to calculate the dimensions of a rectangle given its area and a relationship between its sides is useful in many real-world scenarios. For example, if you're designing a rectangular garden and know the total area you want it to cover, as well as a constraint on the relationship between the length and width (perhaps due to space limitations or aesthetic preferences), you can use these principles to determine the exact dimensions of your garden. This ensures you maximize the use of your space while meeting your design criteria. Let's say you want a garden with an area of 320 square feet, and you want the length to be 4 feet longer than the width. By setting up the equations as we did, you can find the precise length and width needed to achieve your desired garden size and shape. The length is 20 feet and the width is 16 feet.
The width of the jewel case is 16 cm and the length is 20 cm. This is determined through setting up an equation based on the dimensions and solving a quadratic equation. The calculations confirm that the area is correct and the length exceeds the width by 4 cm.
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