Chef Potato's body has a cylinder in the middle and two half spheres that make up his bottom and head. He has the two dimensional shape measurements as follows: 20 cm in total length, 5 cm between the end of the cylinder and the top of the top sphere. Reduce Chef
1.) Potato by a scale factor of 0.5.
2.) Chef Potato needs to be converted from a two dimensional shape to a three dimensional object and then enlarged to 300% of his original size. The three dimensional chef potato is similar in shape to a capsule made of a half sphere at the top, and a half sphere at the bottom and a cylinder in between. Determine the surface area of the enlarged three dimensional chef potato and the scale factor.
1) just multiply each dimension by 0.5
2) Huh? cylinders and spheres are already 3D
Anyway, by enlarging by a factor of 3, its area is enlarged by a factor of 3^2 = 9.
Its current area is 2π*5*10 + 4π*5^2, so just multiply that by 9.
To solve both of these problems, we need to understand the formulas for the surface area of each shape involved.
1.) For the original Chef Potato, we have a cylinder and two half spheres.
To find the surface area of the cylinder, we use the formula A = 2πrh, where r is the radius and h is the height. In this case, the height is the total length of the cylinder, so h = 20 cm. The radius of the cylinder is half of the diameter, which is given as 5 cm. Therefore, r = 5 cm / 2 = 2.5 cm. Plugging these values into the formula, we get A_cylinder = 2π(2.5 cm)(20 cm).
The surface area of each half sphere is given by the formula A = 2πr^2, where r is the radius. In this case, we have two half spheres, one for the bottom and one for the top. The radius of each half sphere is half of the diameter, which is given as 5 cm. Therefore, r = 5 cm / 2 = 2.5 cm. Plugging this value into the formula, we get A_half_sphere = 2π(2.5 cm)^2.
To find the total surface area of the original Chef Potato, we add up the surface areas of the cylinder and the two half spheres: A_total = A_cylinder + A_half_sphere + A_half_sphere.
Now, to reduce Chef Potato by a scale factor of 0.5, we need to multiply the surface area by the square of the scale factor. The scale factor is 0.5, so the reduced surface area is (0.5)^2 * A_total.
2.) For the enlarged three-dimensional Chef Potato, we have a capsule shape made of a half sphere at the top and the bottom, and a cylinder in between. The scale factor for the enlargement is 300%, which means the object will be three times its original size.
To find the surface area of the enlarged three-dimensional Chef Potato, we need to calculate the surface area of the two half spheres and the cylinder, and then add them up.
The formula for the surface area of the cylinder remains the same: A = 2πrh. The radius of the cylinder remains the same, but the height changes due to the enlargement. To find the new height, we multiply the original height by the scale factor of 3. So the height of the enlarged cylinder is 3 * 20 cm = 60 cm. Plugging this value into the formula, we get A_cylinder_enlarged = 2π(2.5 cm)(60 cm).
For the half spheres, the formula for the surface area remains the same: A = 2πr^2. The radius of each half sphere remains the same. Plugging this value into the formula, we get A_half_sphere_enlarged = 2π(2.5 cm)^2.
To find the total surface area of the enlarged three-dimensional Chef Potato, we add up the surface areas of the cylinder and the two half spheres: A_total_enlarged = A_cylinder_enlarged + A_half_sphere_enlarged + A_half_sphere_enlarged.
The scale factor for the enlargement is 300%. This means the new surface area is 3 * A_total.