Answer to Question #198220 in Calculus for John

Question

You have a piece of wire of 10 cm. You use the wire to form either a square or a

circle, or you cut the wire and form a square and a circle .If you cut and use a wire

of length a to form the square and the wire of length 10 - a to form the circle,

the area of the square and/or circle is given by the function

f(a) = (1/16) *a^2+ (1/4pi) (10 - a)^2 = 0.142*a^2 - 1. 592*a + 7.958, 0 <= a <= 10.

For what value of a will the area be a maximum?

2. The function P = f(t) = 300 - t square root of (100 - 2t) gives the weight (in mg) of a population

of bacteria t hours after the start of an experiment.

The domain of the function is [0, 40]

2.1 Find the critical points of the function if f prime of (t) = 3*t - 100/

square root of (100 - 2*t)

.

2.2 Use a number line to find and describe the local extremes of the function.

Answer 1

1.

$f'(a)=0.284a-1.592=0$

Local extremum of f(a):

$a=1.592/0.284=5.6$

$f(5.6)=3.5,f(0)=7.958,f(10)=6.24$

So, the area will be a maximum for $a=0$

2.1

$f'(t)=\frac{3t-100}{\sqrt{100-2t}}=0$

critical point at $t=100/3$ :

$f(100/3)=300-\frac{100}{3}\sqrt{\frac{100}{3}}=300-\frac{1000}{3}\sqrt{\frac{1}{3}}$

2.2

At point $(100/3,300-\frac{1000}{3}\sqrt{\frac{1}{3}})$ the function has local minimum