An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length, l, that is twice the width, w. The volume of the tank is 420 cubic feet. What is the width of the tank to the nearest tenth of a foot? Hint: There is another way to represent the length of the tank in terms other than l.

Question

An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length,  l, that is twice the width, w. The volume of the tank is 420 cubic feet. What is the width of the tank to the nearest tenth of a foot? Hint: There is another way to represent the length of the tank in terms other than  l.

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Given Information: We are given that the volume of the tank is 420 cubic feet and that the length is twice the width. Let's denote the width as w and the length as 2w. The height of the tank is not given, so we'll denote it as h. The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width, and h is the height.Volume Equation: Since we know the volume (V) is 420 cubic feet, we can write the equation 420 = (2w) * w * h, which simplifies to 420 = 2w^2 * h.Height in Terms of Width: We need to find the width (w), but we have two variables in the equation. However, we can express the height (h) in terms of the width (w) and the volume (V). Let's rearrange the equation to solve for h: h = 420 / (2w^2).Finding Width: We don't have enough information to find the exact value of h, but we don't need it to find the width (w). We can assume that the height (h) is such that it will give us a whole number for the width (w) when we solve the equation. This is because the problem does not provide any information about the height, and we are only asked to find the width.Factors of 420: To find the width (w), we need to find the value of w that satisfies the equation 420 = 2w^2 * h. Since we don't have the value of h, we can't solve this equation directly. However, we can look for factors of 420 that are perfect squares, as 2w^2 must be a factor of 420 for the equation to hold true.Checking Factors: The factors of 420 that are perfect squares are 1, 4, and 16. We can divide 420 by each of these factors and see if the resulting quotient is an even number, which would correspond to 2w^2.Identifying Correct Factor: Dividing 420 by 4, we get 105, which is not an even number, so 4 is not the correct factor. Dividing 420 by 16, we get 26.25, which is not an even number either. Therefore, 16 is not the correct factor. However, when we divide 420 by 1, we get 420, which is an even number. This suggests that 2w^2 could be 420.Solving for Width: If 2w^2 is 420, then w^2 is 210. We can now take the square root of both sides to solve for w: w = √210.Calculating Width: Calculating the square root of 210, we get w ≈ 14.49 feet. Rounding to the nearest tenth of a foot, we get w ≈ 14.5 feet.

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