Acertain magical substance that is used to make solid magical spheres costs $500 per cubic foot. the power of a magical sphere depends on its surface area, and a magical sphere can be sold for $20 per square foot of surface area. if you are manufacturing such a sphere, what size should you make them to maximize your profit per sphere? justify why it is a maximum rather than a minimum.

r= 0.08 ft

Step-by-step explanation:

C= $500 per cubic foot

S=$20 per square foot of surface area

We know that

Profit = Selling price - Manufacturing price

P = S - C

Lets take r is the radius of sphere

Surface area ,A= 4πrr²

Volume ,V= 4πr³/3

P = S - C

P = 20 x 4πr² - 500 x 4πr³/3

For maximize the profit

dP/dr = 0

P = 20 x 4πr² - 500 x 4πr³/3

dP/dr = 160 πr - 2000 πr²

160 πr - 2000 πr² = 0

160 - 2000 r = 0

r= 0.08 ft

d²P/dr² = - 4000 πr

d²P/dr²  is negative at all value of r that is why at r=0.08 ft ,it is maximum.

50

Step-by-step explanation:

The translation of the question is as shown;

First note that the original number is 35

The sum of 35 and one-fifth part of itself .

= 35 + 1/5(35)

= 35+7

= 42 ... A

sum of one-seventh part of itself and 3 will be;

= 1/7(35) + 3

= 5+3

= 8... B

Adding A and B together wiill give us the final result

A+B = 42+8

A+B = 50

The sum of 35 and one-fifth part of itself added to the sum of one-seventh

part of itself and 3 is therefore equivalent to 50