There are 32 students in a class. 12 of those students are girls what percent of the class are boys?
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<h3>The possibilities of length and width of the rectangle are </h3><h3>x=1, y=0.24;</h3><h3>x=0.5, y=0.48;</h3><h3>x=0.25, y=0.96;</h3><h3>x=2, y=0.12</h3>Step-by-step explanation:
Given that the area is 0.24 square meter
The area of a rectangle is given by
square units
Let x be the length and y be the width.
Since the area is 0.24 square meter, we have the equation:
, with x and y measures in meters
If we want to know some possibilities of x and y, we can assume a value for one of them, and then calculate the other one using the equation.
Now choosing some values for "x", we have:
Put x = 1
∴ y = 0.24
Now put x = 0.5 we get
∴ y = 0.48
Put x = 0.25
∴ y = 0.96
Put x = 2
∴ y = 0.12
We're given the implicit equation
x² + 2x + y² - 4y = 12
where "implicit" means that one of the variables is considered a function of the other. In this case, the variable y = y(x) depends on the value of the variable x - in other words, the possible values of y are implied by the value of x. We then say that this equation defines y as an implicit function of x.
We're also given the implicit derivative,
dy/dx = (-2x - 2)/(2y - 4)
which tells us the rate of change y as a function of both x and y. The value of dy/dx corresponds to the slope of the tangent line to the curve defined by the implicit equation above at some point (x, y).
The tangent line is horizontal when its slope is zero. This happens when
(-2x - 2)/(2y - 4) = 0
If y ≠ 2, then we can eliminate the denominator and we're left with
-2x - 2 = 0
Solve for x :
-2x = 2
x = -1
This tells us that the points on the curve with x-coordinate -1 have a tangent line to the curve that is horizontal.
The tangent line is vertical when its slope is undefined/infinite, or equivalently when the denominator of dy/dx is zero:
2y - 4 = 0
Solve for y :
2y = 4
y = 2
So any point on the curve with y-coordinate 2 will have a tangent line there that is vertical.
By completing the square in the implicit equation, we can easily identify where these points are located.
x² + 2x + y² - 4y = 12
x² + 2x + 1 + y² - 4y + 4 = 17
(x + 1)² + (y - 2)² = 17
This equation defines a circle centered at (-1, 2) with radius √17. When x = -1, we have
(y - 2)² = 17 ⇒ y = 2 ± √17
and when y = 2, we have
(x + 1)² = 17 ⇒ x = -1 ± √17
See the attached plot to see the circle and the tangents at these points.
Answer:
below
Step-by-step explanation:
steps are shown above
