The rate of change in the number of miles of road cleared per hour by a snowplow is inversely proportional to the depth of the snow. Given that 24 miles per hour are cleared when the depth of the snow is 2.2 inches and 13 miles per hour are cleared when the depth of the snow is 7 inches, then how many miles of road will be cleared each hour when the depth of the snow is 12 inches? (Round your answer to three decimal places.)
m = miles
d = depth of snow
m*d = k
So, you want m such that
m*12 = 24*2.2 = 13*7
Now you have a problem, since 24*2.2 is not equal to 13*7. Maybe your calculation should be based on some average of the two...
To solve this problem, we can use the concept of inverse variation.
Inverse variation states that if two quantities, such as the rate of change and the depth of snow, are inversely proportional, their product remains constant. Mathematically, this can be represented by the equation:
\[ \text{rate} \times \text{depth} = \text{constant} \]
We are given that 24 miles per hour are cleared when the depth of snow is 2.2 inches, and 13 miles per hour are cleared when the depth of snow is 7 inches. Let's use these values to find the constant in the equation.
\[ \text{rate} \times \text{depth} = \text{constant} \]
When the rate is 24 miles per hour and the depth is 2.2 inches:
\[ 24 \times 2.2 = \text{constant} \]
\[ 52.8 = \text{constant} \]
Now that we have the constant, we can use it to find the rate when the depth of snow is 12 inches.
\[ \text{rate} \times 12 = 52.8 \]
\[ \text{rate} = \frac{52.8}{12} \]
\[ \text{rate} \approx 4.400 \text{ miles per hour} \]
Therefore, when the depth of snow is 12 inches, approximately 4.400 miles of road will be cleared each hour.