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Luda [366]
3 years ago
15

The figure here shows triangle AOC inscribed in the region cut from the parabola y=x^2 by the line y=a^2. Find the limit of the

ratio of the area of the triangle to the area of the parabolic region as a approaches zero.

Mathematics
1 answer:
aleksandrvk [35]3 years ago
3 0
Area of the parabolic region = Integral of [a^2 - x^2 ]dx | from - a to a =

(a^2)x - (x^3)/3 | from - a to a = (a^2)(a) - (a^3)/3 - (a^2)(-a) + (-a^3)/3 =

= 2a^3 - 2(a^3)/3 = [4/3](a^3)

Area of the triangle = [1/2]base*height = [1/2](2a)(a)^2 = <span>a^3

ratio area of the triangle / area of the parabolic region = a^3 / {[4/3](a^3)} =

Limit of </span><span><span>a^3 / {[4/3](a^3)} </span>as a -> 0 = 1 /(4/3) = 4/3
</span>
 



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Find the ordered pair that makes both the equations below true
NNADVOKAT [17]

Answer:

(3, 1)

Step-by-step explanation:

(a) Algebraic solution

(1)  y = -⅔x + 3

(2) y =   2x  - 5

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-⅔x + 3 = 2x - 5

Multiply each side by 3

-2x + 9 = 6x - 15

Add 15 to each side

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(b) Graphical solution

In the diagram below, the red line is the graph of Equation (1). The blue line is the graph of Equation (2). The point of intersection is at (3, 1).

7 0
3 years ago
Given an equilateral triangle that has an area of 1959 cm2. What is the length of its sides?
Nina [5.8K]
Equilateral
equi means equal
lateral means sides

all sides are equal
triangle has 3 sides

perimiter=side1+side2+side3
side1=side2=side3


area=1/2 times base times height
base=side
draw a line down the middle of te tiangle so you get 2 right triangles side by side
1/2base is one leg
base is hypotonuse
height is unknown
equilateral means that it is a 30-60-90 triangle with  hypotonuse equal to 2x, height=x√3, bottom is x
1/2 times base times height=1959
1/2 times x times x√3=1959
multiply both sides by 2
(x^2)√3=3918
divide both sides by √3
x^2=3918/√3
square root both sides
x=47.5611
side=hypotonuse=2x
2x=2*47.5611
side=95.12222


legnth of each side is 2\sqrt{ \frac{3918}{ \sqrt{3} } }cm or about 95.12222 cm
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