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Where should the point P be chosen on line segment AB so as to maximize the angle θ? (Assume a = 4 units, b = 5 units, and c = 9

taurus [48]

From the figure, let the distance of point P from point A on line segment AB be x and let the angle opposite side a be M and the angle opposite side c be N.

Using pythagoras theorem,
$\tan M= \frac{a}{b-x} \\ \\ M=\tan^{-1}\left(\frac{a}{b-x}\right)$
and
$\tan N= \frac{c}{x} \\ \\ N=\tan^{-1}\left(\frac{c}{x}\right)$

Angle θ is given by
$\theta=180-M-N \\ \\ =180-\tan^{-1}\left(\frac{a}{b-x}\right)-\tan^{-1}\left(\frac{c}{x}\right)$

Given that a = 4 units, b = 5 units, and c = 9 units, thus
$\theta=180-\tan^{-1}\left(\frac{4}{5-x}\right)-\tan^{-1}\left(\frac{9}{x}\right)$

To maximixe angle θ, the differentiation of <span>θ with respect to x must be equal to zero.
i.e.
$\frac{d\theta}{dx} = -\frac{4}{x^2-10x+41} + \frac{9}{x^2+81} =0 \\ \\ -4(x^2+81)+9(x^2-10x+41)=0 \\ \\ -4x^2-324+9x^2-90x+369=0 \\ \\ 5x^2-90x+45=0 \\ \\ x^2-18x+9=0 \\ \\ x=9\pm6 \sqrt{2}$

Given that x is a point on line segment AB, this means that x is a positive number less than 5.

Thus
$x=9-6 \sqrt{2}=0.5147$

Therefore, The distance from A of point P, so that </span>angle θ is maximum is 0.51 to two decimal places.

6 0

2 years ago

A simple random sample of size nequals15 is drawn from a population that is normally distributed. The sample mean is found to be

Bond [772]

Answer with explanation:

To test the Significance of the population which is Normally Distributed we will use the following Formula Called Z test

$z=\frac{\Bar X - \mu}{\frac{\sigma}{n}}$

$\Bar X =31.1\\\\ \sigma=6.3\\\\ \mu=25\\\\n=15\\\\z=\frac{31.1-25}{\frac{6.3}{\sqrt{15}}}\\\\z=\frac{6.1\times\sqrt{15}}{6.3}\\\\z=\frac{6.1 \times 3.88}{6.3}\\\\z=3.756$

→p(Probability) Value when ,z=3.756 is equal to= 0.99992=0.9999

⇒Significance Level (α)=0.01

We will do Hypothesis testing to check whether population mean is different from 25 at the alpha equals 0.01 level of significance.

→0.9999 > 0.01

→p value > α

With a z value of 3.75, it is only 3.75% chance that ,mean will be different from 25.

So,we conclude that results are not significant.So,at 0.01 level of significance population mean will not be different from 25.

3 0

2 years ago

Think about how to find the area of a rectangle. A=LW and a triangle 1/2B*H. Create a problem using each formula.

WITCHER [35]

Given expressions:

Area of a rectangle = L W

Area of a triangle = $\frac{1}{2} B H$

Problem;

Create problems using the formula;

Solution:

Problem 1; Find the area of a rectangle whose length is 10cm and the width is half the size of the length?

Given parameters:

Length of rectangle = 10cm

Width of rectangle = $\frac{1}{2}$ x length = $\frac{1}{2}$ x 10 = 5cm

So, the area of the rectangle = 10cm x 5cm = 50cm²

2. A triangle with a base length of 30cm and a height of 2cm will have an area of what?

Given parameters:

base length = 30cm

height = 2cm

Solution:

Area of a triangle = $\frac{1}{2}$ x b x h

Area of a triangle = $\frac{1}{2}$ x 30 x 2 = 30cm²

The area of the triangle will be 30cm²

6 0

2 years ago

What’s the rational number has -0.875 as its decimal equivalent

Fynjy0 [20]

Get rid of the decimal first.
-0.875*1000
-875/1000=-0.875
Simplify it since both denominator and numerator is divisible by 5
1/5(-875/1000)
-175/200
Still divisible by 5
1/5(-175/200)
-35/40
Keep going
-7/8
There, now it can’t be more simplified.

8 0

2 years ago

Which expression is equivalent to 16x^15?

JulijaS [17]

Answer is b
As powers add when base is same

3 0

2 years ago

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