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vekshin1
3 years ago
9

Zadanie optymalizacyjne z matematyki. Proszę o rozwiązanie i wyjaśnienie

Mathematics
1 answer:
Yakvenalex [24]3 years ago
5 0

Volume of the pyramid:

V=\dfrac{s^2h}3

Perimeter of the cross-section:

40=\sqrt2\,s+2\sqrt{\dfrac{s^2}2+h^2}=\sqrt2\left(s+\sqrt{s^2+2h^2}\right)

\implies h=\sqrt{\dfrac{(20\sqrt2-s)^2-s^2}2}=\sqrt{400-20\sqrt2\,s}

Area of the cross-section:

P=\dfrac12(\sqrt2\,s)h=\dfrac{sh}{\sqrt2}

\implies P=\dfrac{s\sqrt{400-20\sqrt2\,s}}{\sqrt2}=s\sqrt{200-10\sqrt2\,s}

First derivative test:

\dfrac{\mathrm dP}{\mathrm ds}=\dfrac{20\sqrt2-3s}{\sqrt{4-\dfrac{\sqrt2}5s}}=0\implies s=\dfrac{20\sqrt2}3

Then the height of the cross-section/pyramid is

h=\sqrt{400-20\sqrt2\,s}=\dfrac{20}{\sqrt3}

The volume of the pyramid that maximizes the cross-sectional area P is

V=\dfrac{\left(\frac{20\sqrt2}3\right)^2\frac{20}{\sqrt3}}3=\dfrac{16000}{27\sqrt3}

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What are the coordinates of the endpoints of the mid segment of triangle ABC that is parallel to line AB?
MrMuchimi

Answer:

(4.5,4)  and  (4.5,6.5)

Step-by-step explanation:

we have the coordinates

A(2,7),B(2,2),C(7,6)

we know that

The mid segment of triangle ABC that is parallel to line AB is located between the mid point AC and the mid point BC

The formula to calculate the midpoint between two points is equal to

(\frac{x1+x2}{2},\frac{y1+y2}{2})

step 1

Find the mid point AC

we have

A(2,7),C(7,6)

substitute in the formula

(\frac{2+7}{2},\frac{7+6}{2})

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step 2

Find the mid point BC

we have

B(2,2),C(7,6)

substitute in the formula

(\frac{2+7}{2},\frac{2+6}{2})

(4.5,4)

therefore

The endpoints of the mid segment of triangle ABC that is parallel to line AB are (4.5,4)  and  (4.5,6.5)

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1/19

Step-by-step explanation:

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Step-by-step explanation:

Solution,

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