Q. 75.0( 1 Vote )

# The height of a c

Given the height of a cone increases by k%.

Let x be the height of the cone and Δx be the change in the value of x.

Hence, we have Δx = 0.01kx

Let us assume the radius, the slant height and the semi-vertical angle of the cone to be r, l and α respectively as shown in the figure below. From the above figure, using trigonometry, we have  r = x tan(α)

We also have   l = x sec(α)

(i) The total surface area of the cone is given by

S = πr2 + πrl

From above, we have r = x tan(α) and l = x sec(α).

S = π(x tan(α))2 + π(x tan(α))(x sec(α))

S = πx2tan2α + πx2tan(α)sec(α)

S = πx2tan(α)[tan(α) + sec(α)]

On differentiating S with respect to x, we get  We know   Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as Here, and Δx = 0.01kx

ΔS = (2πxtan(α)[tan(α) + sec(α)])(0.01kx)

ΔS = 0.02kπx2tan(α)[tan(α) + sec(α)]

The percentage increase in S is,  Increase = 0.02k × 100%

Increase = 2k%

Thus, the approximate increase in the total surface area of the cone is 2k%.

(ii) The volume of the cone is given by From above, we have r = x tan(α).   On differentiating V with respect to x, we get  We know   Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as Here, and Δx = 0.01kx

ΔV = (πx2tan2α)(0.01kx)

ΔV = 0.01kπx3tan2α

The percentage increase in V is,   Increase = 0.03k × 100%

Increase = 3k%

Thus, the approximate increase in the volume of the cone is 3k%.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

If y = sin x and RD Sharma - Volume 1

Find the percentaRD Sharma - Volume 1

A circular metal RD Sharma - Volume 1

The radius of a sRD Sharma - Volume 1

The height of a cRD Sharma - Volume 1

Find the approximMathematics - Exemplar

Find the point onMathematics - Board Papers