Mathematics, 24.07.2020 09:01 lberries08

A cone-shaped paper drinking cup is to be made to hold 33 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.) height cm radius cm

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Answer from: princessdriayp6a2o8

The height and the radius of the cylinder are 3.67 centimeters and 5.19 centimeters, respectively.

Step-by-step explanation:

The volume () and the surface area ( $A_{s}$ ) of the cone, measured in cubic centimeters and square centimeters, respectively, are modelled after these formulas:

Volume

$V = \frac{h\cdot r^{2}}{3}$

Surface area

$A_{s} = \pi\cdot r \cdot \sqrt{r^{2}+h^{2}}$

Where:

- Height of the cylinder, measured in centimeters.

- Radius of the base of the cylinder, measured in centimeters.

The volume of the paper drinking cup is known and first and second derivatives of the surface area functions must be found to determine the critical values such that surface area is an absolute minimum. The height as a function of volume and radius of the cylinder is:

$r = \sqrt{\frac{3\cdot V}{h} }$

Now, the surface area function is expanded and simplified:

$A_{s} = \pi\cdot \sqrt{\frac{3\cdot V}{h} }\cdot \sqrt{\frac{3\cdot V}{h}+ h^{2}}$

$A_{s} = \pi\cdot \sqrt{\frac{9\cdot V^{2}}{h^{2}} + 3\cdot V\cdot h }$

$A_{s} = \pi\cdot \sqrt{3\cdot V} \cdot\sqrt{\frac{3\cdot V+ h^{3}}{h^{2}} }$

$A_{s} = \pi\cdot \sqrt{3\cdot V}\cdot \left(\frac{\sqrt{3\cdot V + h^{3}}}{h}\right)$

If $V = 33\,cm^{3}$ , then:

$A_{s} = 31.258\cdot \left(\frac{\sqrt{99+h^{3}}}{h} \right)$

The first and second derivatives of this function are require to determine the critical values that follow to a minimum amount of paper:

First derivative

$A'_{s} = 31.258\cdot \left[\frac{\left(\frac{3\cdot h^{2}}{\sqrt{99+h^{2}}}\right)\cdot h - \sqrt{99+h^{3}} }{h^{2}}\right]$

$A'_{s} = 31.258\cdot \left(\frac{3\cdot h^{3}-99-h^{3}}{h^{2}\cdot \sqrt{99+h^{2}}} \right)$

$A'_{s} = 31.258\cdot \left(\frac{2\cdot h^{3}-99}{h^{2}\cdot \sqrt{99+h^{2}}} \right)$

$A'_{s} = 31.258\cdot \left[2\cdot h\cdot (99+h^{2}})^{-0.5} -99\cdot h^{-2}\cdot (99+h^{2})^{-0.5}\right]$

$A'_{s} = 31.258\cdot (2\cdot h - 99\cdot h^{-2})\cdot (99+h)^{-0.5}$

Second derivative

$A''_{s} = 31.258\cdot \left[(2+198\cdot h^{-3})\cdot (99+h)^{-0.5}-0.5\cdot (2\cdot h - 99\cdot h^{-2})\cdot (99+h)^{-1.5}\right]$

Let equalize the first derivative to zero and solve the resultant expression:

$31.258\cdot (2\cdot h - 99\cdot h^{-2})\cdot (99+h)^{-0.5} = 0$

$2\cdot h - 99 \cdot h^{-2} = 0$

$2\cdot h^{3} - 99 = 0$

$h= \sqrt[3]{\frac{99}{2} }$

$h \approx 3.672\,cm$

Now, the second derivative is evaluated at the critical point:

$A''_{s} = 31.258\cdot \{[2+198\cdot (3.672)^{-3}]\cdot (99+3.672)^{-0.5}-0.5\cdot [2\cdot (3.672) - 99\cdot (3.672)^{-2}]\cdot (99+3.672)^{-1.5}\}$

$A''_{s} = 18.506$

According to the Second Derivative Test, this critical value leads to an absolute since its second derivative is positive.

The radius of the cylinder is: ( $V = 33\,cm^{3}$ and $h \approx 3.672\,cm$ )

$r = \sqrt{\frac{3\cdot V}{h} }$

$r = \sqrt{\frac{3\cdot (33\,cm^{3})}{3.672\,cm} }$

$r \approx 5.192\,cm$

The height and the radius of the cylinder are 3.672 centimeters and 5.192 centimeters, respectively.

Answer from: Quest

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step-by-step explanation:

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Answer from: Quest

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

mark brainliest!

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A cone-shaped paper drinking cup is to be made to hold 33 cm3 of water. Find the height and radius o...