The depth of the snow on Cam's driveway is increasing at a rate of \( r(t)=\frac{t+1}{2} \) centimeters per hour (where \( t \) is the time in hours). At time \( t=5 \), the depth of the snow is 9 centimeters. By how much does the depth of the snow increase between hours 5 and 8 ? Which expression can we use to solve the problem? Choose 1 answer: (A) \( 9+\int_{5}^{8} r(t) d t \) (B) \( r^{\prime}(8) \) (C) \( \int_{5}^{8} r(t) d t \) (D) \( r^{\prime}(8)-9 \) (8)

To find the increase in the depth of the snow between hours 5 and 8, we need to set up an integral of the rate function \( r(t) \) over that time interval. So, to calculate the total increase in snow depth from \( t=5 \) to \( t=8 \), we can use the integral of \( r(t) \) from 5 to 8. The correct expression would be \( \int_{5}^{8} r(t) dt \), which corresponds to option (C). Now, let’s dig into some fun facts! Snow depth measurement has been around since ancient times, with historical records dating back to the early 1700s when scientists first began to document snowfall and understand its impacts on the environment and agriculture. Imagine how much snow has been reported over the centuries! Speaking of snow, did you know that you can apply concepts like this in real life to plan your driveway shoveling schedule effectively? Keeping track of snowfall rates helps you decide when to clear your driveway—for instance, if you see the rate increasing, it might be a good idea to shovel before it becomes a hefty burden. Planning ahead can spare you from being buried under a mountain of snow!