/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 Find the general formula for the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 58

Find the general formula for the surface area of a cone with height \(h\) and base radius \(a\) (excluding the base).

Short Answer

Expert verified
Answer: The general formula for the lateral surface area of a cone with height h and base radius a, excluding the base, is given by: \[A = \pi a \sqrt {a^2 + h^2}\]

Step by step solution

01

Understand the problem

We need to find a general formula to calculate the lateral surface area of a cone, excluding the base. We are given the height (\(h\)) and the base radius (\(a\)). We will find a formula using the slant height of the cone.
02

Determine the slant height

We can notice that there is a right triangle in the cone: the base radius, the height, and the slant height are three sides of the right triangle. To find the slant height, we will apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (in this case, the slant height) is equal to the sum of the squares of the other two side lengths. So, given \(a\) and \(h\), we can determine the slant height \(l\) using this formula: \[l = \sqrt {a^2 + h^2}\]
03

Apply the formula for lateral surface area

Now, we have the slant height \(l\) and base radius \(a\), and we can use the formula for calculating the lateral surface area of a cone. The lateral surface area of a cone is given by: \[A = \pi a l\]
04

Substitute the slant height formula into the lateral surface area formula

We can substitute the expression for \(l\) from Step 2 into the formula for \(A\) from Step 3: \[A = \pi a \sqrt {a^2 + h^2}\]
05

Finalize the general formula

So the general formula for the lateral surface area of a cone with height \(h\) and base radius \(a\) is: \[A = \pi a \sqrt {a^2 + h^2}\] This formula will give us the lateral surface area of the cone, excluding the base.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(p\) is a real number. Let \(S\) consist of the spheres \(A\) and \(B\) centered at the origin with radii \(0

Consider the potential function \(\varphi(x, y, z)=G(\rho),\) where \(G\) is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}} ;\) therefore, \(G\) depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\) b. Let \(S\) be the sphere of radius \(a\) centered at the origin and let \(D\) be the region enclosed by \(S\). Show that the flux of \(\mathbf{F}\) across \(S\) is $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2} G^{\prime}(a) $$ c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use part (c) to show that the flux across \(S\) (as given in part (b)) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.)

Let \(S\) be the paraboloid \(z=a\left(1-x^{2}-y^{2}\right),\) for \(z \geq 0,\) where \(a>0\) is a real number. Let \(\mathbf{F}=\langle x-y, y+z, z-x\rangle .\) For what value(s) of \(a\) (if any) does \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) have its maximum value?

For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. $$\mathbf{v}=\langle-2 z, 0,1\rangle$$

Consider the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}=\frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}},\) where \(p>1\) (the inverse square law corresponds to \(p=3\) ). Let \(C\) be the line from (1,1,1) to \((a, a, a),\) where \(a>1,\) given by \(\mathbf{r}(t)=\langle t, t, t\rangle,\) for \(1 \leq t \leq a\) a. Find the work done in moving an object along \(C\) with \(p=2\) b. If \(a \rightarrow \infty\) in part (a), is the work finite? c. Find the work done in moving an object moving along \(C\) with \(p=4.\) d. If \(a \rightarrow \infty\) in part (c), is the work finite? e. Find the work done in moving an object moving along \(C\) for any \(p>1\) f. If \(a \rightarrow \infty\) in part (e), for what values of \(p\) is the work finite?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.