Find the exact value of cos 45º in simplest form with a rational denominator.
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A conic section is a curve obtained by the intersection of the surface of a cone with a plane. A conic section can be a circle, a hyperbola, a parabola, and an ellipse.
For a circle, the general equation of a circle with center, (a, b), and a radius, r, is of the form
For a hyperbola, the general equation of a hyperbola with center (h, k), and a and b half the lengths of the major and the minor axis respectively is of the form.
For a parabola, the general equation of a parabola with center (h, k), and a multiplier a is of the form
For an ellipse, the general equation of an ellipse with center (h, k), and a and b half the lengths of the major and the minor axis respectively.
Given the equation
It can be rewritten as
This gives an equation of a hyperbola with center (0, 0), half the length of the major axis = 30 and half the length of the minor segment = 10.
The domain of the equation is all real values of x.
For a circle, the general equation of a circle with center, (a, b), and a radius, r, is of the form
For a hyperbola, the general equation of a hyperbola with center (h, k), and a and b half the lengths of the major and the minor axis respectively is of the form.
For a parabola, the general equation of a parabola with center (h, k), and a multiplier a is of the form
For an ellipse, the general equation of an ellipse with center (h, k), and a and b half the lengths of the major and the minor axis respectively.
Given the equation
It can be rewritten as
This gives an equation of a hyperbola with center (0, 0), half the length of the major axis = 30 and half the length of the minor segment = 10.
The domain of the equation is all real values of x.
Answer: x = 5*sqrt(10)
"sqrt" stands for "square root"
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Refer to the attached image. The figure is not to scale. Notice how I've added in y and z (blue and red respectively)
The length y is the bottom horizontal leg for the triangle on the right.
The length z is the bottom horizontal leg for the triangle on the left.
To find x, we need to find y first, which will help us find z, and then we can finally get to x.
Let's find y first
To do this, we use the pythagorean theorem
a^2 + b^2 = c^2
y^2 + 9^2 = 15^2
y^2 + 81 = 225
y^2 + 81-81 = 225-81
y^2 = 144
sqrt(y^2) = sqrt(144)
y = 12 ... keep in mind that y is a length, so it needs to be positive
Notice how the y and z lengths combine to form a total length of 25 units
Therefore,
y+z = 25
12+z = 25
12+z-12 = 25-12
z = 13
Now that we know z = 13, we can find x. Again we use the pythagorean theorem one last time
a^2 + b^2 = c^2
z^2 + 9^2 = x^2
13^2 + 9^2 = x^2
169 + 81 = x^2
250 = x^2
x^2 = 250
sqrt(x^2) = sqrt(250)
x = sqrt(250)
x = sqrt(25*10)
x = sqrt(25)*sqrt(10)
x = 5*sqrt(10)
This is in simplest radical form as we can't factor 10 any further (there are no perfect square factors that go into 10). This is the exact length of side x
Note: using a calculator, 5*sqrt(10) = 15.8113883008419 approximately
"sqrt" stands for "square root"
-------------------------------------------------------
Refer to the attached image. The figure is not to scale. Notice how I've added in y and z (blue and red respectively)
The length y is the bottom horizontal leg for the triangle on the right.
The length z is the bottom horizontal leg for the triangle on the left.
To find x, we need to find y first, which will help us find z, and then we can finally get to x.
Let's find y first
To do this, we use the pythagorean theorem
a^2 + b^2 = c^2
y^2 + 9^2 = 15^2
y^2 + 81 = 225
y^2 + 81-81 = 225-81
y^2 = 144
sqrt(y^2) = sqrt(144)
y = 12 ... keep in mind that y is a length, so it needs to be positive
Notice how the y and z lengths combine to form a total length of 25 units
Therefore,
y+z = 25
12+z = 25
12+z-12 = 25-12
z = 13
Now that we know z = 13, we can find x. Again we use the pythagorean theorem one last time
a^2 + b^2 = c^2
z^2 + 9^2 = x^2
13^2 + 9^2 = x^2
169 + 81 = x^2
250 = x^2
x^2 = 250
sqrt(x^2) = sqrt(250)
x = sqrt(250)
x = sqrt(25*10)
x = sqrt(25)*sqrt(10)
x = 5*sqrt(10)
This is in simplest radical form as we can't factor 10 any further (there are no perfect square factors that go into 10). This is the exact length of side x
Note: using a calculator, 5*sqrt(10) = 15.8113883008419 approximately