# Derivatives Problems

• May 21st 2008, 01:48 PM
lemontea
Derivatives Problems
1) A hot air balloon rises vertically from the ground so that its height after t sec is h= 1/2t^2+1/2t ft (0<equal to t< equal to 60)

a) what is the height of the baloon at the end of 40 sec?
b) What is the average velocity of the balloon between t=0 and t=40
c) What is the velocity of the balloon at the end of 40 sec

2)The quarerly profit (in thousands of dollars) of Cunningham realty is given by
P(x)=-1/3x^2 +7x+30 (0<equal to x < equal to 50)

where x (in thousands of dollars) is the amount of moeny Cunningham spends on advertising per quarter

a) Find P' (x)
b) What is the rate of change of Cunningham's quarterly profit if the amount it spends on advertising is $10,000/quarter (x=10) and$30,000/quarter (x=30)

I am having trouble with derivatives...Can anyone help me with these two problems! GREATLY appreciate it!
• May 21st 2008, 01:59 PM
topsquark
Quote:

Originally Posted by lemontea
A hot air balloon rises vertically from the ground so that its height after t sec is h= 1/2t^2+1/2t ft (0<equal to t< equal to 60)

a) what is the height of the baloon at the end of 40 sec?
b) What is the average velocity of the balloon between t=0 and t=40
c) What is the velocity of the balloon at the end of 40 sec

The velocity of an object is the first time derivative of the position function. So
$v = \frac{dh}{dt} = \frac{1}{2} \cdot 2t^{2 - 1} + \frac{1}{2} \cdot 1 t^{1 - 1} = t + \frac{1}{2}$

The average velocity can be found using the mean value theorem for integrals:
$\bar{v} = \frac{1}{40 - 0} \int_0^{40} v(t)~dt$

$= \frac{1}{40} \left ( \frac{1}{2}t^2 + \frac{1}{2}t \right )|_0^{40}$

and you can take it from there.

-Dan