Question

A ski resort uses a snow machine to control the snow level on a ski slope. Over a 24-hour period the volume of snow added to the slope per hour is modeled by s(t) =24-tsin²(t/14).The rate at which snow melts is modeled by M(t)=10+8cos( t/3)Both S(t) and M(t) have units of cubic yards per hour and t is measured in hours for 0≤t≤24. At time t=0, the slope holds 50 cubic yards of snow. a) Compute the total volume of snow added to the mountain in the first 6-hour period.

Answer 1

To find the total volume of snow added to the mountain in the first 6-hour period, we need to take the definite integral of the snow function over the given time interval.

Step-by-step explanation:

To compute the total volume of snow added to the mountain in the first 6-hour period, we need to find the definite integral of the snow function, s(t), over the interval [0, 6].

Given s(t) = 24 - t*sin^2(t/14), we can integrate s(t) with respect to t using the power rule and the basic integral rule for sin^2(t) to find:

∫(24 - t*sin^2(t/14)) dt = 24t - (7/2)sin(t/7)cos(t/7) + C

Plugging in the values for the definite integral, we have:

∫06(24 - t*sin^2(t/14)) dt = 6*24 - (7/2)sin(6/7)cos(6/7) - 0*24 + (7/2)sin(0/7)cos(0/7)

Simplifying this expression will give us the total volume of snow added in the first 6-hour period.