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 1000 lb. ?
thank you!!
If the linear dimension scales by x,
surface scales by x^2
volume scales by x^3
so, since the weight(or volume) scaled by 1000/64 = (10/4)^3
the surface scales by (10/4)^2 = 6.25
6.25*340 = 2125
To find the surface area of a similar polyhedron made of the same material and weighing 1000 lb, we can use the concept of similarity.
Since the two polyhedra are similar, their corresponding sides are proportional. Let's say the area of the total surface of the original polyhedron is A1 and the weight is W1. The area of the total surface of the similar polyhedron is A2 and the weight is W2.
We are given:
A1 = 340 sq.in.
W1 = 64 lb
W2 = 1000 lb
To find A2, we need to find the ratio of the weights of the polyhedra and use it to scale the surface area.
Weight ratio = W2/W1 = 1000 lb / 64 lb
Surface area ratio = (Weight ratio)^(2/3)
A2 = A1 * (Surface area ratio)
Substituting the given values, we have:
A2 = 340 sq.in. * (1000 lb / 64 lb)^(2/3)
Calculate the value of (1000 lb / 64 lb)^(2/3) first.
(1000 lb / 64 lb)^(2/3) = 15.625^(2/3) ≈ 6.25
Now, substitute this value back into the equation:
A2 = 340 sq.in. * 6.25
A2 ≈ 2125 sq.in.
Therefore, the surface area of the similar polyhedron made of the same material and weighing 1000 lb is approximately 2125 sq.in.
To find the surface area of a similar polyhedron made of the same material and weighing 1000 lb., we need to use the concept of similarity ratios.
Since the two polyhedra are similar, the ratio of their corresponding sides is the same. Let's assume this ratio is 'k'.
The weight of the first polyhedron is 64 lb, and its surface area is 340 sq. in.
Now, let's find the ratio of their weights:
Weight ratio = (Weight of the second polyhedron) / (Weight of the first polyhedron)
Weight ratio = 1000 lb / 64 lb
Next, we need to find the ratio of the surface areas using the weight ratio. Since weight is directly proportional to the surface area (assuming the same material), we can use this relationship to find the ratio of their surface areas.
Surface area ratio = (Surface area of the second polyhedron) / (Surface area of the first polyhedron)
Surface area ratio = Weight ratio
Now, we can set up and solve the proportion to find the unknown ratio, 'k':
Surface area ratio = (Surface area of the second polyhedron) / (Surface area of the first polyhedron)
Weight ratio = (Weight of the second polyhedron) / (Weight of the first polyhedron)
Weight ratio = Surface area ratio
1000 lb / 64 lb = (Surface area of the second polyhedron) / 340 sq. in.
We can rearrange the equation to solve for the unknown surface area:
(Surface area of the second polyhedron) = (1000 lb / 64 lb) * 340 sq. in.
To get the final answer, we can simplify and calculate the value:
(Surface area of the second polyhedron) = 5312.5 sq. in.
Therefore, the surface area of a similar polyhedron made of the same material and weighing 1000 lb is 5312.5 square inches.