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Nikitich [7]
3 years ago
15

The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x^2. Cross sections of the solid perpendi

cular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?
My teacher isn't helpful at all and I'm starting to fail tests, please help
Mathematics
1 answer:
olga55 [171]3 years ago
7 0
He area of an equilateral triangle of side "s" is s^2*sqrt(3)/4. So the volume of the slices in your problem is 

<span>(x - x^2)^2 * sqrt(3)/4. </span>
<span>Integrating from x = 0 to x = 1, we have </span>
<span>[(1/3)x^3 - (1/2)x^4 + (1/5)x^5]*sqrt(3)/4 </span>
<span>= (1/30)*sqrt(3)/4 = sqrt(3)/120 = about 0.0144. </span>

<span>Since this seems quite small, it makes sense to ask what the base area might be...integral from 0 to 1 of (x - x^2) dx = (1/2) - (1/3) = 1/6. Yes, OK, the max height of the triangles occurs where x - x^2 = 1/4, and most of the triangles are quite a bit shorter...


</span>
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Step-by-step explanation:

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zmey [24]

Law of cosines :

The law of cosines establishes:

c ^ 2 = a ^ 2 + b ^ 2 - 2*a*b*cosC.

general guidelines:

The law of cosines is used to find the missing parts of an oblique triangle (not rectangle) when either the two-sided measurements and the included angle measure are known (SAS) or the lengths of the three sides (SSS) are known.


Law of the sines:


In ΔABC is an oblique triangle with sides a, b, and c, then:

\frac{a}{sinA} =\frac{b}{sinB} =\frac{c}{sinC}

The law of the sines is the relation between the sides and angles of triangles not rectangles (obliques). It simply states that the ratio of the length of one side of a triangle to the sine of the angle opposite to that side is equal for all sides and angles in a given triangle.

General guidelines:

To use the law of the sines you need to know either two angles and one side of the triangle (AAS or ASA) or two sides and an opposite angle of one of them (SSA).


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(1) The triangle does not exist.


(2) Two different triangles exist.


(3) Exactly a triangle exists.


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3 years ago
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Answer:

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Step-by-step explanation:

Lisa apparently failed to realize that both terms inside the first set of parentheses have 3x as a factor. They are like terms, so could be combined directly. If Lisa really wants to factor out 3 or 3x, she could do so and then combine the remaining factors at another step.

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