Linda has prepared two cakes. One is in a rectangular pan that measures 9 inches by 13 inches, with a depth of 3 inches. The other is in a circular tube pan which is 12 inches in diameter and 5.50 inches deep. The circular pan which has a cylinder in the centre produces a round cake with a 3-inch diameter hole in the centre.
Linda wants to calculate the cost of making and decorating the circular cake. She will put a uniform layer of icing on the outside and inside walls, and the top of the cake. For decoration, a chocolate string will go on the top circumference of her cake, including enough to go around the centre opening. Calculate the total cost to make and decorate Linda's cake. Show your work.
Batter $0.04 per in.³
Chocolate string $0.39 per ft. (sold in one foot lengths)
Icing $ .01 per in.^ 2 (uniform layer)
Please help me, thank you...
I cant figure it out
Please help
circular tube pan which is 12 inches in diameter and 5.50 inches deep with a 3-inch diameter hole
Just use your formulas for area and volume of a cylinder, and circumference of a circle.
The circumference of a circle is πd
The lateral area of a cylinder is πdh
The area of a circle is πr^2
The volume of a cylinder is πr^2 h
So the area on the outside of the round cake is π*12*5.5 = 66π
and the area on the inside is π*3*5.5 = 16.5π
The area of a circle is πr^2, so the area of the top is π(6^2 - 1.5^2) = 33.75π
The volume of the cake is π(R^2-r^2)h = 33.75*5.5π = 185.625π
So the cost of the cake is
batter: 0.04 * 185.625π = $23.33
icing: 0.01 * (66π+16.5π+33.75π) = $3.65
string: 0.39*(12+3)π = $18.38
To find the volume of each cake, we need to use the formulas for the volume of a rectangular prism and the volume of a cylindrical tube with a hole in the center.
For the rectangular cake, the volume can be calculated as:
Volume = Length x Width x Depth
Length = 9 inches
Width = 13 inches
Depth = 3 inches
Volume = 9 inches x 13 inches x 3 inches
Volume = 351 cubic inches
So, the rectangular cake has a volume of 351 cubic inches.
For the cylindrical cake, we need to subtract the volume of the hole from the total volume of the cylinder.
The total volume of the cylinder can be calculated as:
Volume = π x (radius^2) x height
Diameter = 12 inches
Radius = Diameter / 2 = 12 inches / 2 = 6 inches
Height = 5.5 inches
Volume = π x (6 inches)^2 x 5.5 inches
Volume = π x 36 square inches x 5.5 inches
Volume = 198 square inches x 5.5 inches
Volume = 1089 cubic inches
Now, we need to subtract the volume of the hole. The hole has a diameter of 3 inches, so its radius is 3 inches / 2 = 1.5 inches.
The volume of the hole can be calculated as:
Volume of hole = π x (radius^2) x height
Radius = 1.5 inches
Height = 5.5 inches
Volume of hole = π x (1.5 inches)^2 x 5.5 inches
Volume of hole = π x 2.25 square inches x 5.5 inches
Volume of hole = 12.375 square inches x 5.5 inches
Volume of hole = 67.8125 cubic inches
To find the volume of the cylindrical cake, we subtract the volume of the hole from the total volume of the cylinder:
Volume of cylindrical cake = Total volume - Volume of hole
Volume of cylindrical cake = 1089 cubic inches - 67.8125 cubic inches
Volume of cylindrical cake = 1021.1875 cubic inches
So, the cylindrical cake has a volume of approximately 1021.19 cubic inches.
Thus, Linda has prepared a rectangular cake with a volume of 351 cubic inches and a cylindrical cake with a volume of approximately 1021.19 cubic inches.