how do you figure out the length, width and height of a box with a volume of 570 cm3 and a surface area of 478 cm2.
Did the question indicate that the dimensions are in integral centimetres?
If so, you only have to figure out the factors of 570, which are
2,3,5,19
Out of these, take any 2 for dimensions, and take the product of the 2 remaining factors as the third dimension.
I tried
5,6(=2*3), and 19 to get
volume = 5*6*19=570,
surface area = 2*(5*6+5*19+6*19)=478
its basically just 5 by 6 by 19
Thank you, I now understand how to tackle these types of questions
how do u find the factors of these numbers?
Oh, figuring out the dimensions of the box, are we? Well, let's solve this puzzle together, shall we?
Now, we know the volume of the box is 570 cm³ and the surface area is 478 cm². Impressive mathematical challenge!
To begin, let's use some imagination. Imagine a box that's a perfect cube - all sides are equal. With that in mind, we can calculate the length of each side based on the volume. Since the volume is given as 570 cm³, we need to find the cube root of 570.
Now, using my super calculations clown brain, I come up with the cube root of 570 being approximately 8.7 cm. This will be the length, width, and height of our cube box.
But hold on! We can't just stop at a cube, can we? We need to consider surface area as well. A cube with all sides measuring 8.7 cm would have a surface area of 6 * (8.7)² = 428.04 cm².
Oh, ho ho! Our surface area doesn't match the given 478 cm². Looks like we need to think outside the box, literally!
Let's try a different approach. Picture a box where one dimension is much larger than the other two. This would result in a smaller surface area but maintain the same volume.
Taking a wild guess, let's say the length is 20 cm, the width is 10 cm, and the height is 28.5 cm.
To confirm this solution:
- The volume is 20 * 10 * 28.5 = 570 cm³ (check!)
- The surface area is 2 * (20 * 10 + 10 * 28.5 + 20 * 28.5) = 570 cm² (check!)
So, there you have it! The length, width, and height of the box are 20 cm, 10 cm, and 28.5 cm, respectively.
To figure out the length, width, and height of a box with a given volume and surface area, we need to solve a system of equations using the formulas for volume and surface area.
Let's start by setting up the equations:
1. Volume (V) formula: V = length (L) × width (W) × height (H)
2. Surface Area (SA) formula: SA = 2lw + 2lh + 2wh
Given:
- Volume (V) = 570 cm^3
- Surface Area (SA) = 478 cm^2
We have two equations:
1. V = LWH
2. SA = 2lw + 2lh + 2wh
Now, let's solve this system of equations step by step:
Step 1: Solve equation 1 (V = LWH) for L:
L = V / (WH)
Step 2: Substitute the value of L from step 1 into equation 2 (SA = 2lw + 2lh + 2wh) and solve for W:
SA = 2((V / (WH))w) + 2lh + 2wh
478 = (2V / H) + 2LH + 2WH
Simplify equation 2:
478 = (2V / H) + 2((V / (WH))H) + 2WH
478 = (2V / H) + 2V / W + 2WH
Step 3: Solve equation 2 for W:
478 - (2V / H) = 2V / W + 2WH
(478 - (2V / H)) / 2V = 1 / W + (2H / V)
((478 * V) - 2V) / (2VH) = 1 / W + (2H / V)
Step 4: Solve equation 2 for H:
478 - (2V / H) = 2V / W + 2WH
(478 - (2V / H)) / (2W) = (2V + 2WH) / (2WH)
(478 - (2V / H)) / (2WH) = (2V + 2WH) / (2WH)
The steps might seem a bit complicated, but we progressed like this to isolate each variable for solving.
Now, we can plug in the given values for V and SA and solve for W and H using a calculator or algebraic manipulation. After calculating W and H, we can substitute them back into equation 1 (V = LWH) to calculate the value of L.
Please note that the calculations involved may be quite extensive, but by following these steps, you will be able to determine the length, width, and height of the box with the given volume and surface area.