Two sandboxes with the same area are shown.
The equation [tex]w(3 w+1)=5^2[/tex] represents the area of Sandbox 2 in terms of its width.
Which is the approximate length of the longest side of Sandbox 2? Round the answer to the nearest hundredth of a meter.
A. 2.72 meters
B. 3.06 meters
C. 9.16 meters
D. 10.18 meters
Answer (2)
Solve the quadratic equation 3 w 2 + w − 25 = 0 using the quadratic formula.
Find the positive root: w = 6 − 1 + 301 ≈ 2.7248 .
Calculate the longest side: 3 w + 1 ≈ 3 ( 2.7248 ) + 1 ≈ 9.1744 .
Round to the nearest hundredth: 9.17
Explanation
Understanding the Problem We are given that two sandboxes have the same area. Sandbox 1 is a square with side length 5 m, so its area is 5 2 = 25 m 2 . Sandbox 2 has width w and length 3 w + 1 , and its area is given by the equation w ( 3 w + 1 ) = 5 2 = 25 . We need to find the length of the longest side of Sandbox 2, which is 3 w + 1 , rounded to the nearest hundredth of a meter.
Setting up the Quadratic Equation First, we need to solve the equation w ( 3 w + 1 ) = 25 for w . This equation can be rewritten as 3 w 2 + w = 25 , or 3 w 2 + w − 25 = 0 . This is a quadratic equation in the form a w 2 + b w + c = 0 , where a = 3 , b = 1 , and c = − 25 .
Applying the Quadratic Formula We can use the quadratic formula to find the value(s) of w : w = 2 a − b ± b 2 − 4 a c . Plugging in the values for a , b , and c , we get: w = 2 ( 3 ) − 1 ± 1 2 − 4 ( 3 ) ( − 25 ) = 6 − 1 ± 1 + 300 = 6 − 1 ± 301 .
Calculating the Width Since w must be positive (as it represents the width of a sandbox), we take the positive root: w = 6 − 1 + 301 . Now we can calculate the approximate value of w .
Approximating the Width Using a calculator, we find that 301 ≈ 17.349 . Therefore, w ≈ 6 − 1 + 17.349 = 6 16.349 ≈ 2.7248 .
Calculating the Longest Side Now we can calculate the length of the longest side, which is 3 w + 1 . 3 w + 1 ≈ 3 ( 2.7248 ) + 1 ≈ 8.1744 + 1 ≈ 9.1744 .
Final Answer Rounding the result to the nearest hundredth of a meter, we get 9.17 meters.
Conclusion Therefore, the approximate length of the longest side of Sandbox 2 is 9.17 meters.
Examples
Understanding quadratic equations is crucial in various real-world applications, such as optimizing the design of structures. For instance, engineers use quadratic equations to calculate the optimal dimensions of a bridge's arch to ensure maximum strength and stability while minimizing material usage. Similarly, architects apply these principles to design parabolic structures that efficiently distribute weight and withstand external forces, ensuring the safety and longevity of buildings.
The approximate length of the longest side of Sandbox 2 is calculated to be 9.17 meters. This is determined by solving the quadratic equation for width and then using it to find the length. The final rounded answer to the nearest hundredth is 9.17 meters.
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