Application of Derivatives

The total surface area of a cone is
A=πr2+πrl, where r is radius and l is the slant height.
‌=π(7)2+π(7)⋅√72+72‌‌[∵l=√r2+h2‌ And ‌r=7,h=7]
‌=49π+49√2π=49π(1+√2)
But, the error in measuring r and h is 0.002×7=0.014 feet
So, dr=0.014 and dh=0.014
Now, error in l, denoted by dl and differentiate it w.r.t r and h, we get
dl=‌

∂l

∂r

dr+‌

∂l

∂h

⋅dh
But l=√r2+h2
So, ‌

∂l

∂r

=‌

1

2

(r2+h2)−1∕2⋅2r=‌

r

√r2+h2

=‌

r

l

And ‌

∂l

∂h

=‌

1

2

(r2+h2)−1∕2⋅2h=‌

h

√r2+h2

=‌

h

l

So, ‌‌dl=‌

r

l

dr+‌

h

l

dh=‌

7

7√2

⋅(0.014)+‌

7

7√2

(0.014)
=‌

0.014

√2

+‌

0.014

√2

=‌

2

√2

(0.014)=√2(0.014)
The error in surface area, is dA, i.e.,
dA=‌

∂A

∂r

dr+‌

∂A

∂l

dl
Since, A=πr2+πrl
‌∴‌‌‌

∂A

∂r

=2πr+πl‌ and ‌‌

∂A

∂l

=πr
‌dA=(2πr+πl)⋅dr+(πr)⋅dl
‌=(2πr+πl)(0⋅0.14)+(πr)(0.014)
‌=(14π+7√2π)(0.014)+7√2π(0.014)
‌=14π(0.014)+7√2π(0.014)+7√2π(0.014)
‌=0.196π+0.098√2π+0.098√2π
‌=0.196π+0.196√2π
‌=0.196π(1+√2)
‌=(0.616)(√2+1)