A sodium chloride crystal in the shape of a cube is expanding at the rate of 60 cubic microns per second. How fast is the side of the cube growing when the volume is 1000 cubic microns?

vangstosiis
2022-07-22
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Dominique Ferrell

Answered 2022-07-23
Author has **18** answers

The sodium chloride is a cube.

$V={s}^{3}$

Take the derivative with respect to time, t.

$\frac{dV}{dt}=3{s}^{2}\frac{ds}{dt}$

We know that:

$\frac{dV}{dt}=60$

We want to find $\frac{ds}{dt}$ when V=1000, which means that s=10.

$60=3{\left(10\right)}^{2}\frac{ds}{dt}$

$60=300\frac{ds}{dt}$

$\frac{ds}{dt}=\frac{1}{5}$ microns/second

When the cube has a volume of 1000 $\text{microns}}^{3$, the side of the cube is growing at a rate of $\frac{1}{5}$ $\text{microns/second}$.

$V={s}^{3}$

Take the derivative with respect to time, t.

$\frac{dV}{dt}=3{s}^{2}\frac{ds}{dt}$

We know that:

$\frac{dV}{dt}=60$

We want to find $\frac{ds}{dt}$ when V=1000, which means that s=10.

$60=3{\left(10\right)}^{2}\frac{ds}{dt}$

$60=300\frac{ds}{dt}$

$\frac{ds}{dt}=\frac{1}{5}$ microns/second

When the cube has a volume of 1000 $\text{microns}}^{3$, the side of the cube is growing at a rate of $\frac{1}{5}$ $\text{microns/second}$.

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This might be trivial but I am struggling to justify the following simplification.from:

$h({x}^{\ast}+d)-h({x}^{\ast})\ge \sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

to:

$h({x}^{\ast}+d)-h({x}^{\ast})\ge -\underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

Specifically, why is there a negative in front of the maximization?

Note:

I can get behind the fact that

$\sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}\ge \underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|$

provided the jacobian is nonnegative element-wise. But then why add the negative sign?

$h({x}^{\ast}+d)-h({x}^{\ast})\ge \sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

to:

$h({x}^{\ast}+d)-h({x}^{\ast})\ge -\underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

Specifically, why is there a negative in front of the maximization?

Note:

I can get behind the fact that

$\sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}\ge \underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|$

provided the jacobian is nonnegative element-wise. But then why add the negative sign?

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a. In this ANOVA analysis, what are we trying to determine about the three populations they’re taken from?

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d. In an ANOVA test, what does an F test statistic lower than its critical value tell us about the three populations we’re examining?