Problem 55 - Calculus Early Transcendentals

Question

Surface area of a cone A cone with height \(h\) and radius \(r\) has a lateral surface area (the curved surface only, excluding the base) of \(S=\pi r \sqrt{r^{2}+h^{2}}\) a. Estimate the change in the surface area when \(r\) increases from \(r=2.50\) to \(r=2.55\) and \(h\) decreases from \(h=0.60\) to \(h=0.58\) b. When \(r=100\) and \(h=200,\) is the surface area more sensitive to a small change in \(r\) or a small change in \(h ?\) Explain.

Step-by-Step Solution

Answer

Question: Estimate the change in surface area for a cone when its radius changes from 2.50 to 2.55 and its height changes from 0.60 to 0.58. Determine whether the surface area is more sensitive to a change in radius or height for the cone when r = 100 and h = 200. Answer: The estimated change in surface area is approximately 0.443 square units. The surface area of the cone is more sensitive to a change in the radius (r) than a change in the height (h).

Step 1: Calculate the surface area for initial measurements

Using the formula for the surface area, \(S=\pi r \sqrt{r^{2}+h^{2}}\), we will calculate the initial surface area when \(r=2.50\) and \(h=0.60\). \(S = \pi (2.50) \sqrt{(2.50)^{2}+(0.60)^{2}} \approx 16.086\) The initial surface area is approximately 16.086 square units.

Step 2: Calculate the surface area for new measurements

Now, we will calculate the new surface area when \(r=2.55\) and \(h=0.58\) using the same formula: \(S = \pi (2.55) \sqrt{(2.55)^{2}+(0.58)^{2}} \approx 16.529\) The new surface area is approximately 16.529 square units.

Step 3: Estimate the change in surface area

We can estimate the change in surface area by calculating the difference between the new and initial surface areas: \(\Delta S \approx 16.529 - 16.086 \approx 0.443\) The estimated change in surface area is approximately 0.443 square units. #b- Determine surface area sensitivity to r and h#

Step 1: Calculate partial derivatives

To determine the sensitivity of the surface area to changes in radius and height, we will calculate the partial derivatives with respect to \(r\) and \(h\). \(\frac{\partial S}{\partial r} = \pi \left(\sqrt{r^{2}+h^{2}} + \frac{r^2}{\sqrt{r^{2}+h^{2}}}\right)\) \(\frac{\partial S}{\partial h} = \pi r \frac{h}{\sqrt{r^{2}+h^{2}}}\)

Step 2: Evaluate the partial derivatives when r=100 and h=200

Now, we will evaluate the partial derivatives at the given values of \(r=100\) and \(h=200\). \(\frac{\partial S}{\partial r} \Bigr|_{r=100, h=200} \approx \pi \left(\sqrt{100^{2}+200^{2}} + \frac{100^2}{\sqrt{100^{2}+200^{2}}}\right) \approx 314.6\) \(\frac{\partial S}{\partial h} \Bigr|_{r=100, h=200} \approx \pi (100) \frac{200}{\sqrt{100^{2}+200^{2}}} \approx 157.3\)

Step 3: Determine the sensitivity of surface area to r and h

Comparing the values of the partial derivatives, the value for \(\frac{\partial S}{\partial r}\) is larger than the value for \(\frac{\partial S}{\partial h}\). It means that the surface area of the cone is more sensitive to a small change in the radius \(r\) than a small change in the height \(h\).

Key Concepts

Partial Derivatives Simplified

Partial derivatives allow us to understand how a function changes when we tweak one of the variables while keeping the others constant. In our cone example, the surface area depends on both the radius (\(r\)) and the height (\(h\)). When we take partial derivatives, we're essentially zooming in on how changes in just one parameter (either \(r\) or \(h\)) affect the surface area.

• To calculate \(\frac{\partial S}{\partial r}\), we treat \(h\) as a constant and differentiate with respect to \(r\). This gives insight into how sensitive the surface area is to changes in radius.
• Similarly, \(\frac{\partial S}{\partial h}\) examines how the surface changes with alterations in height, holding the radius constant.

Understanding partial derivatives makes it easier to pinpoint which dimensions significantly impact an object’s properties, like change in surface area in our cone.

Understanding Sensitivity Analysis

Sensitivity analysis is an investigation into how the output of a model is affected by changes in input parameters. In our context, we're looking at how variations in the radius and height of the cone affect its surface area.

• Using partial derivatives, we saw that a change in radius has a more significant effect on the cone's surface area compared to a similar change in height. This is because the value of \(\frac{\partial S}{\partial r}\) was larger than \(\frac{\partial S}{\partial h}\).
• This means that if you were to slightly increase the radius versus slightly increasing the height, the surface area changes more with the radius tweak.

Sensitivity analysis is a powerful tool for predicting and optimizing real-life scenarios, allowing for informed decisions based on which variables most influence the desired outcome.

Exploring Cone Geometry

Understanding cone geometry helps us to derive formulas like the one for surface area. A cone is a three-dimensional geometric shape that narrows smoothly from a flat circular base to a point called the apex.

• The slant height, \(\sqrt{r^2 + h^2}\), is crucial as it forms the hypotenuse of a right triangle with the radius and height representing the other two sides.
• The lateral surface area of a cone, provided by \( \pi r \sqrt{r^2+h^2} \), represents just the curved surface without the base.
• Understanding these relationships allows for deeper insights into the mathematical properties and applications of conical shapes.

By grasping cone geometry, we can derive, visualize, and solve problems more effectively involving spatial dimensions and measurements.