Parallelepiped Volume: Find The Edge Lengths

Hey guys! Let's dive into a fun math problem involving a parallelepiped. We're given the volume and the ratio of its dimensions, and our mission is to find the actual lengths of its edges. Buckle up, it's gonna be a mathematical adventure!

Understanding the Problem

So, the problem states that we have a parallelepiped with a volume of 360 cm³. The dimensions are in the ratio of 2:3:5. This means that if we let 'x' be a common factor, the lengths of the edges can be represented as 2x, 3x, and 5x. Our goal is to find the actual values of these edge lengths.

Setting Up the Equation

The volume of a parallelepiped is given by the product of its three dimensions. In our case, that's:

Volume = length * width * height

Or, in terms of our 'x':

360 = (2x) * (3x) * (5x)

Solving for x

Let's simplify and solve for 'x'.

360 = 30x³

Divide both sides by 30:

x³ = 12

Now, take the cube root of both sides:

x = ∛12

Finding the Edge Lengths

Now that we have the value of 'x', we can find the lengths of the edges:

Edge 1: 2x = 2 * ∛12 Edge 2: 3x = 3 * ∛12 Edge 3: 5x = 5 * ∛12

To approximate the cube root of 12, we know that 2³ = 8 and 3³ = 27, so ∛12 is somewhere between 2 and 3. A closer approximation is ∛12 ≈ 2.289.

So, the edge lengths are approximately:

Edge 1: 2 * 2.289 ≈ 4.578 cm Edge 2: 3 * 2.289 ≈ 6.867 cm Edge 3: 5 * 2.289 ≈ 11.445 cm

Analyzing the Answer Choices

Now, let's look at the answer choices provided:

(a) 4, 6, 10 cm (b) 3, 4, 5 cm (c) 6, 9, 15 cm (d) 2, 3, 5 cm (e) 5, 7, 10 cm

Looking at our approximations, option (a) seems the closest when considering some rounding might be involved. However, let's verify if the ratio 2:3:5 holds true for option (a).

Detailed Solution and Verification

Alright, let’s break down this problem step by step to make sure we nail it. We're given that the volume of the parallelepiped is 360 cm³ and that the dimensions are in the ratio 2:3:5. This means we can represent the dimensions as 2x, 3x, and 5x.

The volume of a parallelepiped is calculated by multiplying its three dimensions:

Volume = Length * Width * Height

In our case:

360 = (2x) * (3x) * (5x)

Now, let’s simplify this equation:

360 = 30x³

Divide both sides by 30:

x³ = 12

Now, we need to find the cube root of 12:

x = ∛12

This value of x is crucial because it will help us determine the actual dimensions of the parallelepiped. We can use this to evaluate the answer choices and see which one fits perfectly.

Now, let’s consider the answer choices:

(a) 4, 6, 10 cm (b) 3, 4, 5 cm (c) 6, 9, 15 cm (d) 2, 3, 5 cm (e) 5, 7, 10 cm

Let's test each option to see which one satisfies both the ratio 2:3:5 and the volume of 360 cm³.

Option (a): 4, 6, 10 cm

Let's check if these dimensions are in the ratio 2:3:5. We can write the ratios as:

4:6:10

Divide each number by 2:

2:3:5

So, the dimensions are indeed in the correct ratio. Now let's check the volume:

Volume = 4 * 6 * 10 = 240 cm³

This doesn’t match our given volume of 360 cm³, so option (a) is incorrect.

Option (b): 3, 4, 5 cm

Let's check the ratio:

3:4:5

This is not in the ratio 2:3:5, so option (b) is incorrect.

Option (c): 6, 9, 15 cm

Let's check if these dimensions are in the ratio 2:3:5. We can write the ratios as:

6:9:15

Divide each number by 3:

2:3:5

So, the dimensions are in the correct ratio. Now let's check the volume:

Volume = 6 * 9 * 15 = 810 cm³

This doesn’t match our given volume of 360 cm³, so option (c) is incorrect.

Option (d): 2, 3, 5 cm

Let's check the ratio:

2:3:5

This is the correct ratio. Now let's check the volume:

Volume = 2 * 3 * 5 = 30 cm³

This doesn’t match our given volume of 360 cm³, so option (d) is incorrect.

Option (e): 5, 7, 10 cm

Let's check the ratio:

5:7:10

This is not in the ratio 2:3:5, so option (e) is incorrect.

The Correct Approach

Since none of the provided options seem to directly fit, let's go back to our original equation and solve for x properly.

We have:

360 = (2x)(3x)(5x)

360 = 30x³

Divide by 30:

x³ = 12

So, x = ∛12. Now we find the dimensions:

Length = 2x = 2∛12 Width = 3x = 3∛12 Height = 5x = 5∛12

Now, let's approximate ∛12. We know it's between 2 and 3 (since 2³ = 8 and 3³ = 27). A closer approximation is around 2.289.

Length ≈ 2 * 2.289 ≈ 4.578 cm Width ≈ 3 * 2.289 ≈ 6.867 cm Height ≈ 5 * 2.289 ≈ 11.445 cm

Given these approximations, none of the provided answer choices exactly match. However, if we analyze and consider potential rounding errors or simplifications, let’s re-evaluate.

We made an error in our initial assumption. The problem states the volume is 360 cm³, and the dimensions are in the ratio 2:3:5. Let's represent the dimensions as 2x, 3x, and 5x. Thus:

(2x)(3x)(5x) = 360 30x³ = 360 x³ = 12 x = ∛12

Now, let's reconsider the options, keeping in mind that we might need to find a common factor that gives us the correct ratio and volume.

Looking closely, there's a mistake in the initial evaluation. We need to find x such that the product of the dimensions equals 360.

Given the correct x = ∛12, the dimensions are 2∛12, 3∛12, and 5∛12.

After a detailed review, it appears there was a miscalculation or misunderstanding in the initial approach and evaluation of the answer choices. The correct method involves setting up the equation based on the volume and ratio, solving for x, and then determining the dimensions. Let's re-examine this with a clear, step-by-step approach.

So after careful calculation, none of the options accurately represents the solution. There may be a typo in the original answers or in the question itself. The correct method is outlined above, ensuring the relationship between edge lengths aligns with the volume.

Conclusion

After a thorough analysis, none of the provided answer choices perfectly match the calculated edge lengths based on the given volume and ratio. This suggests there might be a slight error in the answer choices or the problem statement itself. But hey, that's math for ya! Always keep questioning and double-checking! Keep practicing, and you'll nail these problems in no time!***