Math help? 1-2 please and thank you.

Mathematics

1 answer:

daser333 [38]3 years ago

5 0

1 (TRY IT). If a shape has four right angles, it is a square.

2 (TRY IT).
Converse: If you must come to a stop, then the traffic light is red.

Inverse: If the traffic light is not red, then you must not come to a stop.

Contrapositive: If you must not come to a stop, then the traffic light is not red.

I hope these are the answers you are looking for. If not, please message me again and we'll look into it further. Please also message me if my logic is flawed.

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Find the maximum volume of a cylindrical soda can such that the sum of its height and its circumference is 120 cm.

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The volume of such a can with base radius $r$ and height $h$ would be

$V(r,h)=\pi r^2h$

We desire the base's circumference and the can's height to add to 120, i.e.

$2\pi r+h=120\implies h=120-2\pi r$

Substituting this into $V(r,h)$ allows us to reduce the volume to a function of a single variable $r$ :

$V(r,120-2\pi r)=V(r)=\pi r^2(120-2\pi r)=120\pi r^2-2\pi^2r^3$

Taking the derivative with respect to $r$ yields

$V'(r)=240\pi r-6\pi^2r^2$

Set this equal to 0 and find any critical points:

$240\pi r-6\pi^2r^2=0\implies 6\pi r(40-\pi r)=0\implies r=0\text{ or }r=\dfrac{40}\pi$

This suggests the can will have maximum volume when its radius is $\dfrac{40}\pi\approx12.7$ cm, which would give a volume of about 20,371 sq. cm.

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