A toy baseball bat comes with 3 plastic balls in a box that is a rectangular solid. The box is just big enough to hold the 3 balls. The radius of the balls is 1.9 in. What is the volume of the air in the box surrounding the balls?
Consider each ball as a sphere enclosed in a cube. Then the volume of air is the difference in volume between the cube and the sphere, multiplied by 3 because there are 3 plastic balls.
To find the volume of the air in the box surrounding the balls, we first need to find the volume of the box.
Since the box is a rectangular solid, we need to find the length, width, and height of the box. However, we are given that the box is just big enough to hold the 3 balls, which means the width and length of the box will be equal to the diameter of the balls.
Since the diameter of the balls is twice the radius, the diameter of the balls is 2 * 1.9 = 3.8 inches.
Now, let's assume that the length, width, and height of the box are also 3.8 inches.
The volume of a rectangular solid is given by the formula:
Volume = length * width * height
So, the volume of the box is:
Volume of the box = 3.8 in * 3.8 in * 3.8 in
Calculating this, we get:
Volume of the box = 54.872 in³
Now that we have the volume of the box, we need to subtract the volume of the 3 balls from it to find the volume of the air in the box surrounding the balls.
The formula to find the volume of a sphere is:
Volume of a sphere = (4/3) * π * radius³
Since there are 3 balls, the volume of the 3 balls is:
Volume of the 3 balls = 3 * [(4/3) * π * (1.9 in)³]
Calculating this, we get:
Volume of the 3 balls = 77.781 in³ (rounded to three decimal places)
Finally, subtracting the volume of the 3 balls from the volume of the box, we get the volume of the air in the box surrounding the balls:
Volume of the air in the box = Volume of the box - Volume of the 3 balls
Volume of the air in the box = 54.872 in³ - 77.781 in³
Calculating this, we get:
Volume of the air in the box = -22.909 in³ (rounded to three decimal places)
Therefore, the volume of the air in the box surrounding the balls is approximately -22.909 cubic inches.
To find the volume of the air in the box surrounding the balls, we need to calculate the volume of the entire box and subtract the volume of the three balls.
1. Find the volume of each ball:
The formula for the volume of a sphere is given by V = (4/3)πr³, where r is the radius of the sphere.
Let's calculate the volume of one ball:
V₁ = (4/3)π(1.9 in)³ ≈ 21.381 in³
2. Calculate the total volume of the three balls:
Since we have three identical balls, the total volume of all three balls will be 3 times the volume of one ball:
V_total = 3 * V₁ = 3 * 21.381 in³ = 64.143 in³
3. Find the dimensions of the box:
Since the box is a rectangular solid, it will have three dimensions: length (L), width (W), and height (H).
4. Calculate the volume of the box:
The formula for the volume of a rectangular solid is given by V = L * W * H.
We need to find the dimensions of the box, so we have enough space to hold the three balls.
5. Subtract the volume of the three balls from the volume of the box to get the volume of the air surrounding the balls.
Without knowing the dimensions of the box, we cannot provide the exact volume of the air surrounding the balls. If you have the dimensions of the box, please provide them so we can further assist you.