SOLUTION: Please need help with the following problems.Your help is appreciated. 1.The period of pendulum is directly proportional to the square root of its lenght.If the pendulum has a l

Question 74739This question is from textbook algebra and trigonometry with analytic geometry
: Please need help with the following problems.Your help is appreciated.
1.The period of pendulum is directly proportional to the square root of its lenght.If the pendulum has a lenght of 6 feet and a period of 2 seconds,to what lenght should it be shortened to achieve a 1 second period?
2.Express the following statement as a formula with the value of the constant of propotionality determined with the given conditions;w varies directly as x and inversely as the square of y.If x=15 and y=5.then w=36.
3.find the roots of the polynomial x^3-x^2+16x-16.
4.Given that (3x-a)(x-2)(x-7)=3x^3-32x^2+81x-70,determine the value of a.
This question is from textbook algebra and trigonometry with analytic geometry

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
1.
Since the period is directly proportional to the square root of its length the equation looks like

So if L=6 and T=2, we can find k

Now plug in T=1 to find L

Multiply both sides by
Square both sides

So if the pendulum is 4/3 feet then the period is 1 second

2.
Since w varies directly as x, the equation looks like

Since w also varies inversely as the square of y, it further looks like

If x=15, y=5 and w=36 then we can find k

Multiply both sides by 25
Divide both sides by 15

If we plugged in this value of k, the equation would look like:

Factor out x from the first group and 16 from the 2nd
Factor to
Add like terms of (x-1). Note: if we let we get
Notice how we have a product of factors. Set them equal to zero.

There's one root

Plug this into the quadratic formula to solve for x

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

The discriminant -63 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

In the field of imaginary numbers, the square root of -63 is + or - .

The solution is

Here's your graph:

So the roots are:
, , and

4.
foil the first 2 parenthesis
multiply the remaining 2 parenthesis

Set the equations equal to each other
Since the last term so
So the equation is

Note: I don't know if you copied the equation correctly or not but
is equal to not notice the change of 81 to 87 for the x coefficient.
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