The volume of two similar rectangular prisms are 64cm^3 and 1000cm^3. The surface area of the smaller figure is 112cm^2. What is the surface area in square centimeters of the larger figure?
Well, well, well! Looks like we have some math fun ahead of us! Let's crunch those numbers, shall we?
We know that the volume of the smaller figure is 64 cm^3 and its surface area is 112 cm^2. The volume of the larger figure is 1000 cm^3.
Now, here's a little trick that might come in handy: since the figures are similar, the ratio of their volumes will be equal to the ratio of their surface areas, squared.
So let's set up an equation:
(64/1000)^2 = 112/S
We want to find the surface area of the larger figure, so let's call it S. Now, let's do some mathematical magic and solve for S:
(64/1000)^2 = 112/S
64^2/1000^2 = 112/S
4096/1000000 = 112/S
Now, let's cross multiply:
(4096 * S) = (112 * 1000000)
Divide both sides by 4096 to get S by itself:
S = (112 * 1000000) / 4096
Now, you do the rest of the math and voila! You'll find the surface area in square centimeters of the larger figure. Remember, math can be fun too, just like a clown juggling numbers in the circus!
To find the surface area of the larger figure, we can use the concept of ratios. Let's assume the dimensions of the smaller figure are length, width, and height, and the dimensions of the larger figure are L, W, and H.
Since the two figures are similar, the ratio of their volumes is equal to the scale factor cubed:
(Volume of larger figure) / (Volume of smaller figure) = (LWH) / (lwh) = (L/l) * (W/w) * (H/h)
Given that the volume of the smaller figure is 64 cm³ and the volume of the larger figure is 1000 cm³, we can write the equation:
1000 / 64 = (L/l) * (W/w) * (H/h)
We are given that the surface area of the smaller figure is 112 cm². The surface area of a rectangular prism is given by:
Surface area = 2lw + 2lh + 2wh
For the smaller figure, we have:
2lw + 2lh + 2wh = 112
Now, let's find the ratio of surface areas:
(Surface area of larger figure) / (Surface area of smaller figure) = [(LW)/(lw)] + [(LH)/(lh)] + [(WH)/(wh)]
We don't need the specific values of L, W, H, l, w, h. We just need to compute the ratio of the surface areas.
Since the ratios of each pair of corresponding dimensions will be the same, we can rewrite the equation as:
(Surface area of larger figure) / (Surface area of smaller figure) = [(L/l) * (W/w)] + [(L/l) * (H/h)] + [(W/w) * (H/h)]
Now, let's substitute the given information into the equation:
(Surface area of larger figure) / 112 = [(1000/64)^(2/3)] + [(1000/64)^(1/3)] + [(1000/64)^(1/3)]
Simplifying this equation will give us the ratio of the surface areas.
(Surface area of larger figure) = 112 * [(1000/64)^(2/3)] + 2 * 112 * [(1000/64)^(1/3)] * [(1000/64)^(1/3)]
After evaluating this expression, you will find the surface area of the larger figure in square centimeters.
To find the surface area of the larger figure, we can use the concept of similarity between the two rectangular prisms.
Similar figures have corresponding sides that are proportional to each other. In other words, if the scale factor between the two figures is "k," then each side of the larger figure is "k" times the length of the corresponding side of the smaller figure.
Let's set up the ratio of the volumes first:
(volume of larger figure) / (volume of smaller figure) = (length of larger figure)³ / (length of smaller figure)³
Given that the volumes of the two figures are 1000 cm^3 and 64 cm^3, we have:
1000 cm^3 / 64 cm^3 = (length of larger figure)³ / (length of smaller figure)³
Simplifying the above equation, we have:
(length of larger figure)³ = 1000 cm^3 / 64 cm^3 * (length of smaller figure)³
(length of larger figure)³ = 15.625 * (length of smaller figure)³
Now, let's find the scale factor by taking the cube root of both sides:
length of larger figure = (15.625 * (length of smaller figure)³)^(1/3)
length of larger figure = 15.625^(1/3) * (length of smaller figure)^(1/3)
Since the surface area of a rectangular prism depends on the length, width, and height, and we have the ratio of the lengths, we can conclude that the ratio of the surface areas is equal to the ratio of the lengths squared.
(surface area of larger figure) / (surface area of smaller figure) = (length of larger figure)² / (length of smaller figure)²
Plugging in the value we have for the length ratio:
(surface area of larger figure) / (112 cm²) = (15.625^(1/3) * (length of smaller figure)^(1/3))² / (length of smaller figure)²
We know that (length of smaller figure) = ∛(64 cm³) = 4 cm.
Let's substitute this value into the equation:
(surface area of larger figure) / (112 cm²) = (15.625^(1/3) * (4 cm)^(1/3))² / (4 cm)²
Now, we can calculate the value of (surface area of larger figure):
(surface area of larger figure) = (s surface area of smaller figure) * (15.625^(1/3) * (4 cm)^(1/3))² / (4 cm)²
Calculating this expression will give you the surface area of the larger figure in square centimeters.