To determine the velocity [tex]\( v \)[/tex] of a bowling ball after it has fallen 40 feet, we can use the formula:
[tex]\[ v = \sqrt{2 g h} \][/tex]
where:
- [tex]\( g \)[/tex] is the acceleration due to gravity, which is approximately 32 feet per second squared on Earth.
- [tex]\( h \)[/tex] is the height the object has fallen, which in this case is 40 feet.
Let's go through the steps to find [tex]\( v \)[/tex].
1. Substitute the given values for [tex]\( g \)[/tex] and [tex]\( h \)[/tex] into the formula:
[tex]\[ g = 32 \, \text{ft/s}^2 \][/tex]
[tex]\[ h = 40 \, \text{ft} \][/tex]
Thus, the formula becomes:
[tex]\[ v = \sqrt{2 \cdot 32 \cdot 40} \][/tex]
2. Calculate the product inside the square root:
[tex]\[ 2 \cdot 32 = 64 \][/tex]
[tex]\[ 64 \cdot 40 = 2560 \][/tex]
So we have:
[tex]\[ v = \sqrt{2560} \][/tex]
3. Determine the square root of 2560:
[tex]\[ v \approx 50.59644256269407 \][/tex]
4. Round the result to the nearest tenth:
[tex]\[ v \approx 50.6 \][/tex]
Therefore, the velocity of a bowling ball after it has fallen 40 feet, rounded to the nearest tenth, is approximately [tex]\( 50.6 \, \text{ft/s} \)[/tex].
