Find two positive numbers that satisfy the given requirements. The sum of the fi

Question

Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 51 and the product is a maximum. ________ (first number) _________ (second number) Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 76 meters length _____ m width ______ m A farmer plans to enclose a rectangular pasture to river. (see figure). The pasture must contain 45,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river? x = _____ m y = _____ m

Explanation / Answer

1)

let positive numbers be x,y

sum of fiest number squared and the second number is 51

x2+y=51

=>y=51-x2

product ,p=xy

p=x(51-x2)

p=(51x-x3)

for maximum product,dp/dx=0,d2p/dx2 <0

dp/dx=51-3x2,d2p/dx2 =-6x

51-3x2=0

=>x2=17

=>x=17

at x=17 ,d2p/dx2 =-617 <0

x=17, y=51-x2

=>y=51-17=34

two positive numbers are 17 ,34

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2)

let length =x , width =y

perimeter =2(x+y)=76

=>x+y=38

=>y=38-x

area ,A=xy

A=x(38-x)

A=(38x-x2)

for maximum area,dA/dx=0,d2A/dx2 <0

dA/dx=38-2x,d2A/dx2 <-2

38-2x=0

=>x=19

y=38-x ,x=19

=>y=38-19

=>y=19

the dimensions for maximum area are

length =19m

width =19m

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