Layla Melton

2023-03-24

A hemispherical dome of radius 40 feet is to be given 7 coats of paint, each of which is 1/100 inch thick. How to use linear approximation to estimate the volume of paint needed for the job?

Razorel1l1

Beginner2023-03-25Added 6 answers

I'd go with the conclusion that, for a sphere:

$V=\frac{4}{3}\pi {r}^{3}$ and $\frac{dV}{dr}=4\pi {r}^{2}$

[ie the surface area of a sphere of radius $r$ is the derivative wrt $r$ of its volume]

To begin with, we can state that:

$\delta V=\frac{dV}{dr}\delta r=4\pi {r}^{2}\delta r$

and so with $\delta r=7\cdot \frac{1}{12\cdot 100}$ (7 layers, and adjusting to Imperial ft measurements), we have

$\delta V=4\pi {\left(40\right)}^{2}\cdot \frac{7}{100}=\frac{112}{3}\pi \approx {\phantom{\rule{1ex}{0ex}}\text{117.3 ft}}^{3}$

Actual increase is

$\Delta V=\frac{4}{3}\pi ({(40+\frac{7}{1200})}^{3}-{40}^{3})={\phantom{\rule{1ex}{0ex}}\text{117.3 ft}}^{3}$

$V=\frac{4}{3}\pi {r}^{3}$ and $\frac{dV}{dr}=4\pi {r}^{2}$

[ie the surface area of a sphere of radius $r$ is the derivative wrt $r$ of its volume]

To begin with, we can state that:

$\delta V=\frac{dV}{dr}\delta r=4\pi {r}^{2}\delta r$

and so with $\delta r=7\cdot \frac{1}{12\cdot 100}$ (7 layers, and adjusting to Imperial ft measurements), we have

$\delta V=4\pi {\left(40\right)}^{2}\cdot \frac{7}{100}=\frac{112}{3}\pi \approx {\phantom{\rule{1ex}{0ex}}\text{117.3 ft}}^{3}$

Actual increase is

$\Delta V=\frac{4}{3}\pi ({(40+\frac{7}{1200})}^{3}-{40}^{3})={\phantom{\rule{1ex}{0ex}}\text{117.3 ft}}^{3}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?