the area of the total surface of a polyhedron weighing 64 lb. is 340 sq. in. What is the surface of a similar polyhedron made of the same material and weighing 1000lb. ?
the ratio of the volumes (weight) of two similar figures is proportional to the cube of their sides
so the ratio of sides for the two solids
= cuberoot(1000/64) = 2.5
the ratio of the surface are is proportional to the square of their sides
so
x/340 = 2.5^2 / 1^2
x = 340(6.25) = 2125 square inches
To find the surface area of a similar polyhedron made of the same material but weighing 1000 lb, we can use the concept of similarity.
Let's assume the ratio of the weights of the two polyhedra is "w". In this case, the weight of the first polyhedron is 64 lb, and the weight of the second polyhedron is 1000 lb. Therefore, we have:
w1 / w2 = 64 / 1000
To find the surface area ratio, we know that the ratio of the weights of two similar objects is equal to the ratio of the areas of their corresponding sides raised to the power of 2. So:
(w1 / w2) = (A1 / A2)^2
We will use this relationship to find the surface area of the second polyhedron.
Let A1 be the surface area of the first polyhedron (340 sq.in). We can rewrite the equation as:
(64 / 1000) = (340 / A2)^2
Simplifying this equation, we get:
64 / 1000 = 340^2 / A2^2
Cross-multiplying:
64 * A2^2 = 1000 * 340^2
Dividing both sides by 64:
A2^2 = (1000 * 340^2) / 64
Taking the square root of both sides:
A2 = √[(1000 * 340^2) / 64]
Calculating this expression:
A2 ≈ √[(1000 * 115600) / 64] ≈ √(18000000 / 64) ≈ √(281250) ≈ 530.33
Therefore, the surface area of the similar polyhedron made of the same material and weighing 1000 lb is approximately 530.33 square inches.
To find the surface area of a similar polyhedron, you need to know the relationship between the scale of the polyhedra and the relationship between their weights.
Let's assume that the scale factor between the two polyhedra is "k". This means that all corresponding sides are multiplied by "k" to get the dimensions of the larger polyhedron.
We are given that the weight of the first polyhedron is 64 lb and its surface area is 340 sq. in. We also know that the weight of the second, larger polyhedron is 1000 lb and we need to find its surface area.
Step 1: Determine the scale factor (k):
Since weight is directly proportional to volume, and volume is directly proportional to the cube of the scale factor, we can use the weight ratio to find the scale factor.
(weight of larger polyhedron) / (weight of smaller polyhedron) = (k^3)
1000 lb / 64 lb = k^3
k = ∛(1000/64) ≈ 2.587
Step 2: Calculate the surface area of the larger polyhedron:
The surface area of a polyhedron is proportional to the square of the scale factor since all corresponding sides are multiplied by the same scale factor.
(surface area of larger polyhedron) = (surface area of smaller polyhedron) * (k^2)
= 340 sq. in * (2.587^2) ≈ 2170 sq. in
Therefore, the surface area of the similar polyhedron made of the same material and weighing 1000 lb is approximately 2170 sq. in.