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xxTIMURxx [149]

2 years ago

6

The angle of elevation of the top of a radio mast from a point due east of it and 96 m away from its base is 30°. From another p

oint, due west of the mast, the angle of elevation of the top is 60°. Calculate the distance of the second point from the base of the mast.

1 answer:

marshall27 [118]2 years ago

4 0

Answer:

32 m

Step-by-step explanation:

<u>Trigonometric ratios</u>

$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$

where:

$\theta$ is the angle
O is the side opposite the angle
A is the side adjacent the angle
H is the hypotenuse (the side opposite the right angle)

Calculate the height of the radio mast using the tan trig ratio.

Given:

$\theta$ = 30°
O = h (height of radio mast)
A = 96 m

$\implies \sf tan(30^{\circ})=\dfrac{ \sf h}{96}$

$\implies \sf h=96 \tan (30^{\circ})$

$\implies \sf h=\dfrac{96 \sqrt{3}}{3}$

$\implies \sf h=32 \sqrt{3}\:\:m$

To calculate the distance from the second point to the base of the mast, use the tan trig ratio:

Given:

$\theta$ = 60°
O = 32√3 m
A = d

$\implies \sf tan(60^{\circ})=\dfrac{ \sf 32 \sqrt{3}}{d}$

$\implies \sf \sqrt{3}=\dfrac{ \sf 32 \sqrt{3}}{d}$

$\implies \sf d=\dfrac{ \sf 32 \sqrt{3}}{\sqrt{3}}$

$\implies \sf d=32\:\: m$

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Answer:

99% confidence interval for the given specimen is [3.4125 , 3.4155].

Step-by-step explanation:

We are given that a laboratory scale is known to have a standard deviation (sigma) or 0.001 g in repeated weighing. Scale readings in repeated weighing are Normally distributed with mean equal to the true weight of the specimen.

Three weighing of a specimen on this scale give 3.412, 3.416, and 3.414 g.

Firstly, the pivotal quantity for 99% confidence interval for the true mean specimen is given by;

P.Q. = $\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\bar X$ = sample mean weighing of specimen = $\frac{3.412+3.416+3.414}{3}$ = 3.414 g

$\sigma$ = population standard deviation = 0.001 g

n = sample of specimen = 3

$\mu$ = population mean

<em>Here for constructing 99% confidence interval we have used z statistics because we know about population standard deviation (sigma).</em>

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P(-2.5758 < $\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }$ < 2.5758) = 0.99

P( $-2.5758 \times {\frac{\sigma}{\sqrt{n} } }$ < ${\bar X - \mu}$ < $2.5758 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.99

P( $\bar X-2.5758 \times {\frac{\sigma}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.5758 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.99

<u>99% confidence interval for</u> $\mu$ = [ $\bar X-2.5758 \times {\frac{\sigma}{\sqrt{n} } }$ , $\bar X+2.5758 \times {\frac{\sigma}{\sqrt{n} } }$ ]

= [ $3.414-2.5758 \times {\frac{0.001}{\sqrt{3} } }$ , $3.414+2.5758 \times {\frac{0.001}{\sqrt{3} } }$ ]

= [3.4125 , 3.4155]

Therefore, 99% confidence interval for this specimen is [3.4125 , 3.4155].

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+

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4

2

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=

2

Now, selecting one of the points we can use the point slope formula to find the equation.

The point slope formula is:

y

−

y

1

=

m

(

x

−

x

1

)

Substituting one of our points gives:

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−

−

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2

(

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−

0

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y

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x

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