A rectangular prism with a volume of 400 cubic centimeters has the dimensions [tex]$x+1$[/tex] centimeters, [tex]$2x$[/tex] centimeters, and [tex]$x+6$[/tex] centimeters. The equation [tex]$2x^3+14x^2+12x=400$[/tex] can be used to find [tex]$x$[/tex]. What is the length of the longest side? Use a graphing calculator and a system of equations to find the answer.
A. 4 centimeters
B. 5 centimeters
C. 8 centimeters
D. 10 centimeters
Answer (2)
The longest side of the rectangular prism is calculated to be 10 centimeters. After solving the cubic equation, we find x ≈ 4 , which leads to the side lengths of 5 cm, 8 cm, and 10 cm. Therefore, the correct answer is 10 centimeters.
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Simplify the given equation 2 x 3 + 14 x 2 + 12 x = 400 to x 3 + 7 x 2 + 6 x − 200 = 0 .
Find the approximate real root of the cubic equation using a graphing calculator, resulting in x ≈ 4 .
Calculate the lengths of the sides using the value of x : x + 1 = 5 cm, 2 x = 8 cm, and x + 6 = 10 cm.
Identify the longest side among the calculated lengths, which is 10 cm. The final answer is 10
Explanation
Understanding the Problem We are given a rectangular prism with dimensions x + 1 , 2 x , and x + 6 centimeters, and a volume of 400 cubic centimeters. We are also given the equation 2 x 3 + 14 x 2 + 12 x = 400 , which can be used to find x . Our goal is to find the length of the longest side.
Simplifying the Equation First, let's simplify the equation by dividing both sides by 2: x 3 + 7 x 2 + 6 x = 200 Then, rewrite the equation in the standard form of a cubic equation: x 3 + 7 x 2 + 6 x − 200 = 0
Finding the Value of x Now, we need to find the value of x that satisfies this equation. We can use a graphing calculator or a numerical method to find the approximate real root of the equation. Using a graphing calculator, we find that x ≈ 4 .
Calculating the Side Lengths Now that we have the value of x , we can find the lengths of the sides of the rectangular prism: Side 1: x + 1 = 4 + 1 = 5 cm Side 2: 2 x = 2 ( 4 ) = 8 cm Side 3: x + 6 = 4 + 6 = 10 cm
Identifying the Longest Side The lengths of the sides are 5 cm, 8 cm, and 10 cm. The longest side is 10 cm.
Final Answer Therefore, the length of the longest side is 10 centimeters.
Examples
Imagine you are designing a storage container and need to maximize its length while keeping the volume fixed. This problem demonstrates how to find the dimensions of a rectangular prism given its volume, which is crucial in optimizing storage space. By solving for the variable 'x', you can determine the exact measurements needed to achieve the desired volume and ensure the longest side is maximized. This is useful in various real-world applications, such as packaging design, warehouse optimization, and construction planning, where efficient use of space is essential.
