To determine how many loose gala apples can fit in the box, we can start by calculating the volume of the box and the approximate volume of a gala apple.
Step 1: Calculate the volume of the box.
The dimensions of the box are:
- Length = 16 inches
- Width = 9 inches
- Height = 9 inches
The volume \( V_{box} \) can be calculated using the formula for the volume of a rectangular prism:
\[ V_{box} = \text{Length} \times \text{Width} \times \text{Height} \]
\[ V_{box} = 16 , \text{inches} \times 9 , \text{inches} \times 9 , \text{inches} = 1296 , \text{cubic inches} \]
Step 2: Calculate the approximate volume of a gala apple.
The volume \( V_{apple} \) of a sphere can be calculated using the formula:
\[ V_{apple} = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the apple. The diameter of a gala apple is given as 3.5 inches, so the radius \( r \) is:
\[ r = \frac{3.5}{2} = 1.75 , \text{inches} \]
Now, let's calculate the volume of one gala apple:
\[ V_{apple} = \frac{4}{3} \pi (1.75)^3 \]
Calculating \( (1.75)^3 \):
\[ (1.75)^3 \approx 5.359375 \]
Now substituting this into the volume formula:
\[ V_{apple} \approx \frac{4}{3} \pi (5.359375) \approx \frac{4}{3} \times 3.14159 \times 5.359375 \approx 22.407 , \text{cubic inches} \]
Step 3: Calculate the approximate number of apples that can fit in the box.
To find the number of apples that can fit, we divide the volume of the box by the volume of one apple:
\[ \text{Number of apples} = \frac{V_{box}}{V_{apple}} \approx \frac{1296 , \text{cubic inches}}{22.407 , \text{cubic inches}} \approx 57.8 \]
Since you cannot have a fraction of an apple, we round down to the nearest whole number:
\[ \text{Number of apples} \approx 57 \]
Thus, approximately 57 gala apples should fit in the box.
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