Determine the perimeter of a square with a diagonal of √72 centimeters

1 answer:

const2013 [10]3 years ago

7 0

The correct answer is 23.6
First find the length of the square root of 72 which is 8.4
Next divide 8.4 by the square root of to to find the length of a side 5.9
Multiply 5.9 by 4 to get 23.6

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4 years ago

Gas is escaping from a spherical balloon at the rate of 12 ft3/hr. At what rate (in feet per hour) is the radius of the balloon

bija089 [108]

Answer:

This is the rate at which the radius of the balloon is changing when the volume is 300 $ft^3$ $\frac{dr}{dt}=-\frac{3}{225^{\frac{2}{3}}\pi ^{\frac{1}{3}}} \:\frac{ft}{h} \approx -0.05537 \:\frac{ft}{h}$

Step-by-step explanation:

Let $r$ be the radius and $V$ the volume.

We know that the gas is escaping from a spherical balloon at the rate of $\frac{dV}{dt}=-12\:\frac{ft^3}{h}$ because the volume is decreasing, and we want to find $\frac{dr}{dt}$

The two variables are related by the equation

$V=\frac{4}{3}\pi r^3$

taking the derivative of the equation, we get

$\frac{d}{dt}V=\frac{d}{dt}(\frac{4}{3}\pi r^3)\\\\\frac{dV}{dt}=\frac{4}{3}\pi (3r^2)\frac{dr}{dt} \\\\\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

With the help of the formula for the volume of a sphere and the information given, we find $r$

$V=\frac{4}{3}\pi r^3\\\\300=\frac{4}{3}\pi r^3\\\\r^3=\frac{225}{\pi }\\\\r=\sqrt[3]{\frac{225}{\pi }}$

Substitute the values we know and solve for $\frac{dr}{dt}$

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}\\\\\frac{dr}{dt}=\frac{\frac{dV}{dt}}{4\pi r^2} \\\\\frac{dr}{dt}=-\frac{12}{4\pi (\sqrt[3]{\frac{225}{\pi }})^2} \\\\\frac{dr}{dt}=-\frac{3}{\pi \left(\sqrt[3]{\frac{225}{\pi }}\right)^2}\\\\\frac{dr}{dt}=-\frac{3}{\pi \frac{225^{\frac{2}{3}}}{\pi ^{\frac{2}{3}}}}\\\\\frac{dr}{dt}=-\frac{3}{225^{\frac{2}{3}}\pi ^{\frac{1}{3}}} \approx -0.05537 \:\frac{ft}{h}$