To find the radius of the bucket, we can use the formula for the volume of a cylinder, which is:
\[ V = \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height. Given that the volume \( V \) is 606 cubic inches and the height \( h \) is 10.4 inches, we can substitute these values into the formula and solve for \( r \).
First, we substitute the known values into the volume formula:
\[ 606 = 3.14 r^2 (10.4) \]
Next, we simplify the equation:
\[ 606 = 32.456 r^2 \]
Now, we solve for \( r^2 \):
\[ r^2 = \frac{606}{32.456} \]
Calculating the right side:
\[ r^2 \approx 18.66 \]
Now, we take the square root to find \( r \):
\[ r \approx \sqrt{18.66} \approx 4.32 \text{ inches} \]
Thus, the radius of the bucket is approximately 4.32 inches.
Effect of Changing Height on Radius
If the height of the bucket is changed while keeping the volume constant at 606 cubic inches, the radius will change inversely proportional to the height. This is derived from the volume formula.
If we express the radius in terms of height: \[ r^2 = \frac{V}{\pi h} \]
This means: \[ r = \sqrt{\frac{V}{\pi h}} \]
From this equation, if \( V \) is constant and \( h \) increases, then \( \frac{1}{h} \) becomes smaller, causing \( r \) to decrease. Conversely, if \( h \) decreases, then \( \frac{1}{h} \) increases, which causes \( r \) to increase.
Thus, when the height is changed while keeping the volume constant, the radius changes in such a way that increases in height result in a decrease in radius and decreases in height result in an increase in radius.
1 answer