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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.

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Most popular questions from this chapter

Suppose \(y=f(x)\) is a continuous and positive function on \([a, b] .\) Let \(S\) be the surface generated when the graph of \(f\) on \([a, b]\) is revolved about the \(x\) -axis. a. Show that \(S\) is described parametrically by \(\mathbf{r}(u, v)=\langle u, f(u) \cos v, f(u) \sin v\rangle,\) for \(a \leq u \leq b, 0 \leq v \leq 2 \pi\) b. Find an integral that gives the surface area of \(S\) c. Apply the result of part (b) to find the area of the surface generated with \(f(x)=x^{3},\) for \(1 \leq x \leq 2\) d. Apply the result of part (b) to find the area of the surface generated with \(f(x)=\left(25-x^{2}\right)^{1 / 2},\) for \(3 \leq x \leq 4\).

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and parametric description of the sphere.

Evaluate a line integral to show that the work done in moving an object from point \(A\) to point \(B\) in the presence of a constant force \(\mathbf{F}=\langle a, b, c\rangle\) is \(\mathbf{F} \cdot \overrightarrow{A B}\)

Let S be the disk enclosed by the curve \(C: \mathbf{r}(t)=\langle\cos \varphi \cos t, \sin t, \sin \varphi \cos t\rangle,\)for \(0 \leq t \leq 2 \pi,\) where \(0 \leq \varphi \leq \pi / 2\) is a fixed angle. What is the length of \(C ?\)

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T\) (the Laplacian of \(T\)). Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100(1+\sqrt{x^{2}+y^{2}+z^{2}})$$
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