Question

please do not skip and excercise I need all pf them for strudying porpuses

Answer each of the problems. Make sure to show all your work. 1) A pump is used to fill an inflatable mattress at a constant rate of 0.25 cubic feet per second. If the mattress is completely inflated after 1 minute, how much air can the mattress hold? Include units in your answer 2) The same mattress in Problem 1 is later deflated by opening its release valve. Air escapes at a rate of rt) 0.25e 0s where ris in cubic feet per second. Assuming the mattress is still completely inflated when the valve is opened, how much air is released in the first 30 seconds? 3) Refer to the information in Problems 1 and 2. How much air remains in the mattress after 45 seconds from opening the release valve? 4) A projectile travels at a velocity of v(t) 45-32t, where vt) is measured in feet per second. Find the change in position (displacement) of the projectile on the time interval [O, 3]. Include units in your answer. 5) Refer to the scenario in Problem 4. If the projectile started with an initial position at s(0) 25 feet, find its position at t 3. 6) The graph of g'(t) on the interval [0, 10 is shown below. If g(0) -3, find g10) g'(t) 4 10 -4 7) A tumor is found with an initial mass of 3.5 grams. Its growth rate is measured and recorded in the table below, where R(t) is the rate of growth in grams per month. Use a trapezoidal approximation with four subintervals to determine the increase in mass of the tumor over the 4-month period. R( 02 0.3 0.5 0.7 0.8 8) Refer to the information in Problem 7. Find the mass of the tumor when t4. 40 9) A skydiver jumps from a plane from 4000 meters and begins to fal at a velocty of v where vit) is measured in meters per second. What is the skydiver's altitude after 15 seconds? 60 (2t+1 10) The rate of change in the volume of a tank is known to be dV 0.6t cos(0.08f -1, where V) is in dt gallons per minute and Osts 10. If the tank has a volume of 14 gallons initially, what is its volume at 10 minutes

Answer 1

Solution:

1. AIr incoming rate =.25 cubic feet per second

Air capacity in 1 minute = Air incoming rate * time

=.25*60

=15 cubic feet of air

As we know that 1 cubic foot =.0283 cubic metre

hence Air capacity=15*.0283.4245 m^3

2. Air escape rate function r(t)=.25e^-.05t

for t=30 sec

r(30)=.25e^-.05*30

=7.1343 cubic feet

i.e =.0283*7.1343

=.2019 m^3

3 Air full capacity = 15 cubic feet of air

at t=45 sec

r(45)=.25e^-.05*45

=10.7 cubic feet of air

Air remaining = 15-10.7

=4.2987 cubic feet of air

4 velocity function v(t)=45-32t

as we know that velocity is given by

v=ds/dt

thus

ds=v.dt

$\int_{0}^{3}$ds=$\int_{0}^{3}$(45-32t).dt

s= (45t-16t²)

=(45*3-16*3^2)-(45*02-16*0)

=-9 feet

5.

As s(0)=25

s=45t*16t^2+c

thus c=25

hence

s=45t-16t^2+25

at t=3

s=45*3-16*3^2+25

=16 feet