Archive for November, 2014

Find the equation of the parabola determined by the given information.

Find the equation of the parabola determined by the given information.

Focus (3,5), directrix x = -1

 

Solution:-

Notice that the directrix is a vertical line. Since the axis of symmetry is perpendicular to the directrix, then it is a horizontal line.

The parabola with a horizontal axis of symmetry will have directrix x = h –p,

focus (h + p,k), and standard equation (y - k)^{2} = 4p(x – h).

Since the directrix of the parabola is x = -1, then -1, then -1 = h – p.

Since the x-value of the focus is 3, then 3 = h + p.

Find h and  p using any method for solving a  system of equation. Using the elimination method, eliminate p and solve for h.

h  = 1

substitute the value for h into one of the equation to solve for p.

p = 2

k is the y – value for h , p, and k into standard equation of a parabola.

(y - 5)^{2}= 4*2(x – 1)

(y - 5)^{2}= 8(x – 1)

Find the distance between the pair of points.

Find  the distance between the pair of points.

Give an exact answer and an approximation to three decimal places.

(-5,6) and (2, -3)

 

Solution:-

 

To find the distance between two points, use the distance formula below.

d =  \sqrt{( x_{2}- x_{1})^2+( y_{2}- y_{1})^2}.

Substitute the value and calculate.

d =  \sqrt{( 2-(-5))^2+( -3-6)^2}

=\sqrt{(7)^2+(-9)^2}

= \sqrt{130}

This is the exact value of the distance between the two point.

Now find the approximate distance to three decimal places.

\sqrt{130}  \approx 11.402

 

Use the confidence interval to find the estimated margin of error. Then find the sample mean.

Use the confidence interval to find the estimated margin of error. Then find the sample mean.

A biologist reports  a confidence interval of (3.8, 6.4) when estimating the mean height ( in centimeters) of a sample of seedlings.

Solution:-

The  confidence interval is \overline{x}\pm E where \overline{x} is the sample mean and E is the margin of error.

Divide the width of the confidence interval by 2 to find the margin of error.

\frac{6.4-3.8}{2}=1.3

Therefore, the margin of error is 1.3

The sample mean overlibe{1.3} is the middle of the confidence interval. To find the sample mean, either add the margin of error to the left endpoint or subtract the margin of error from the right endpoint. In this problem, add the margin of error to the left endpoint.

3.8 + 1.3 = 5.1

Therefore , the sample mean is 5.1.

Construct the confidence interval for the population mean

 

Construct the confidence interval for the population mean .

c = 0.90, \overline{x}s = 17.8, s = 12.0, and n = 110

Solution

Use the Formula E =  z_{c}\frac{\sigma }{\sqrt{n}}

to find the margin of error, where z_{c} is the z-score corresponding to an area of c,

\sigma is the population standard deviation and

n is the sample size.

When n\geq 30 you can use s in place of \sigma, where s is the sample standard deviation.

z_{c}  = \frac{1}{2}(1-c).

Substitute the value of c and evaluate to find the area in each tail.

 

z_{c}  =  \frac{1}{2}(1-c)

=\frac{1}{2}(1-0.90) = 0.05

Use a standard normal table to find the positive critical value the corresponds to a tail area of 0.05

z_{c} = 1.645

The population standard deviation \sigma is unknown but because n\geq30, use s in place of \sigma.

Substitute the value z_{c} = 1.645,  \sigma  \approx s \approx12.0, and n = 110 to find E.

 

E = z_{c}\frac{\sigma }{\sqrt{n}}

1.645\times \frac{12}{\sqrt{110}}

\approx1.9

Use the margin of error to find the left endpoint.

Left endpoint = \overline{x} – E

= 17.8 – 1.9 = 15.9

Now find the right endpoint

Right endpoint = \overline{x} + E

= 17.8 + 1.9 = 19.7

Therefore,  a  90% confidence interval for \mu is (15.9, 19.7).