Cynthia has leftovers in a rectangular casserole dish that measures 8 in by 12 in by 2 in tall. She estimates that approximately half the dish’s capacity remains.
1. What volume of food is left in the dish?
2. She stores the leftovers in a cylindrical container that measures 5 inches in diameter. What height will the food reach in the new container?
#1. 1/2 (8*12*2) = 96 in^3
#2. πr^2 h = 96
So plug in r = 5/2 and solve for h.
To find the volume of food left in the rectangular casserole dish, we'll first calculate the volume of the dish and then find half of that.
1. Volume of rectangular casserole dish:
The formula for the volume of a rectangular prism is length × width × height. Given that the dish measures 8 in by 12 in by 2 in tall, the volume of the dish is:
Volume = 8 in × 12 in × 2 in = 192 cubic inches
To find half of the volume, we'll divide it by 2:
Half Volume = 192 cubic inches ÷ 2 = 96 cubic inches
Therefore, there are approximately 96 cubic inches of food left in the dish.
2. Height of food in the new cylindrical container:
The formula for the volume of a cylinder is π × r^2 × h, where "r" is the radius of the circular base and "h" is the height. The diameter of the circular base is given as 5 inches, so the radius (r) will be half of that, which is 2.5 inches.
We want to find the value of "h" (height) when the volume of the cylinder is equal to 96 cubic inches.
Volume of cylinder = π × r^2 × h
96 cubic inches = π × (2.5 inches)^2 × h
Simplifying the equation and isolating "h":
h = (96 cubic inches) / (π × (2.5 inches)^2)
Using a calculator, we can calculate the value of "h":
h ≈ 3.88 inches
Therefore, the food will reach a height of approximately 3.88 inches in the new cylindrical container.
To find the volume of food left in the dish, we need to find the volume of the whole dish and then divide it by 2 since only half the capacity remains.
1. The volume of a rectangular prism (casserole dish) is given by the formula:
Volume = Length x Width x Height
Substituting the given values:
Volume = 8 in x 12 in x 2 in
Volume = 192 in³
Since only half the capacity remains, the volume of food left in the dish is:
Volume = 192 in³ / 2
Volume = 96 in³
So, the volume of food left in the dish is 96 cubic inches.
Now, for the second question:
2. The volume of a cylindrical container is given by the formula:
Volume = π x (Radius)² x Height
To find the height of the food in the cylindrical container, we need to know the radius.
The diameter is given as 5 inches, and the radius (r) is half the diameter, so:
Radius (r) = Diameter / 2
Radius (r) = 5 in / 2
Radius (r) = 2.5 in
Now we can find the height:
Volume of food = π x (Radius)² x Height
Substituting the known values:
96 in³ = π x (2.5 in)² x Height
To solve for the height, we can rearrange the equation:
Height = 96 in³ / (π x (2.5 in)²)
Height ≈ 3.87 in
Therefore, the height the food will reach in the new cylindrical container is approximately 3.87 inches.