destinyhope4776

30.04.2021 •

Mathematics

Can someone answer this its just boxplots :)))

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Ответ:

osmarirodriguez1163

19.03.2021

Mathematics

Figure is attached below:

$100.04\pi$

Step-by-step explanation:

Given Data:

triangle vertices= A,B,C=(0,0), (7,0), (0,7)

Required: Find Volume of the solid=V=?

For this type of question we need to use the following formula of slicing method

$\int\limits^a_bA {(x)} \, dx................Eq(1)$

where a and b are limits which shows that solid extends from $x=a\ to\ x=b$ with the known cross section area A(x) perpendicular to x-axis.

From the Figure.2 we can see that limits are from x=0 to x=7 and the red line is the diameter of the given semi circle.

We can see that if x increases then the diameter decreases along the hypotenuse thus diameter of the semi circle =d= 7-x

Radius is half of the diameter, so Radius of semi circle $=r=\frac{7-x}{2}$

Here, Area A(x) is the area of the semi circle

so $A(x)=\frac{1}{2}(\pi ) r^{2}$

Putting values, we get

$A(x)=\frac{1}{2}(\pi ) (\frac{7-x}{2}) ^{2}$

Putting values in Eq(1), we get

$V=\int\limits^7_0 {\frac{1}{2}(\pi ) (\frac{7-x}{2}) ^{2}} \, dx$

$V=\frac{\pi }{2}\int\limits^0_7 {(\frac{7-x}{2} )^{2} } \, dx$

$V=\frac{\pi }{2}\int\limits^0_7 {(\frac{49+x^{2} -2(7)(x)}{4} ) } \, dx$ expanding formula ( $(a+b)^{2}=a^{2}+b^{2}+2ab$ )

$V=\frac{\pi }{8}\int\limits^7_0 ({49+x^{2} -14x)} \, dx$

$V=\frac{\pi }{8}(\int\limits^7_0 {49} \, dx+\int\limits^7_0 {x^{2} } \, dx +\int\limits^7_0 {14x} \, dx )$

$V=\frac{\pi }{8}(49x+\frac{x^3}{3}+\frac{14x^2}{2}) (for\ x=0\ to\ 7)$

Putting x=0 we get everything zero that's why only have expression for value of x=7 as:

$V=\frac{\pi }{8}(49*7+\frac{7^{3}}{3}+14*\frac{7^2}{2} )$

$V=\frac{\pi }{8}(343+114.33+343)$

$V=\frac{\pi }{8}(800.33)\\\\ V=100.04\pi }$

Use the general slicing method to find the volume of the solid whose base is the triangle with verti

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