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sesenic [268]
2 years ago
12

How high is the peak of the Matterhorn? Below is a picture of the Matterhorn in Switzerland. As a surveyor, it is your job to es

timate the elevation of the mountain. Using a device called a theodolite, you measure two angles from the horizontal plane to the top of the mountain as shown below. You also record the distance between the two measurements to be 1500 ft. Your elevation is known to be at 9,756ft above sea level. Determine the height of the Matterhorn (above sea level) to the nearest foot.
Mathematics
1 answer:
Oxana [17]2 years ago
3 0

Answer:

E\approx14,962\text{ feet}

Step-by-step explanation:

Please refer to the attachment below.

As shown, we can split the diagram into a scalene triangle and a right triangle.

The strategy here is to use the Law of Sines to find <em>x</em>, then use right triangle ratios to find <em>h</em>, the height of Matterhorn (with respect to our elevation).

Let’s start with the scalene triangle.

To use the Law of Sines, we will need two angles and the sides opposite to the angles.

Since we need to find <em>x</em>, we need the angle opposite to <em>x</em>. This is already given to us as 30°.

We can use the side that measures 1,500. The angle opposite to it <em>θ</em> is unknown.

So, we will need to determine <em>θ</em>. From supplementary angles, we know that:

\alpha+35=180

So, it follows that:

\alpha=145^\circ

The interior angles of a triangle always sum to 180°. Hence:

30+\alpha+\theta=180

By substitution:

30+145+\theta=180

Hence:

\theta=5^\circ

Now, we can use the Law of Sines, given by:

\displaystyle \frac{\sin(A)}{a}=\frac{\sin(B)}{b}

It is important to align the angles correctly with the sides.

So, we will substitute <em>θ </em>= 5° for A and 1,500 for a.

And we will substitute 30° for B and <em>x</em> for <em>b</em>. Hence:

\displaystyle \frac{\sin(5^\circ)}{1,500}=\frac{\sin(30^\circ)}{x}

Solve for <em>x</em>. Cross-multiply:

x\sin(5^\circ)=1500\sin(30^\circ)

Hence:

\displaystyle x=\frac{1500\sin(30^\circ)}{\sin(5^\circ)}

When doing trigonometry (or any math, really), try not to round intermediate values, as this may result in inaccuracies in the final answer. Hence, we will leave this as is.

Now, we can use the right triangle.

We know that an angle is 35°.

The height <em>h</em> is opposite to our angle.

And <em>x </em>was<em> </em>our hypotenuse. Therefore, we can use the sine ratio given by:

\displaystyle \sin(x^\circ)=\frac{\text{Opposite}}{\text{Hypotenuse}}

In this case, our angle is 35°.

The opposite side is <em>h</em>.

And the hypotenuse is <em>x</em>. Hence, by substitution:

\displaystyle \sin(35^\circ)=\frac{h}{x}

Solve for <em>h</em>. Multiply both sides by <em>x: </em>

h=x\sin(35^\circ)

Since we already know the value of <em>x: </em>

<em />\displaystyle h=\frac{1500\sin(30^\circ)}{\sin(5^\circ)}\cdot\sin(35^\circ)<em />

Simplify. So, the height of Matterhorn with respect to our current elevation is:

\displaystyle h =\frac{1500\sin(30^\circ)\sin(35^\circ)}{\sin(5^\circ)}

The <em>total</em> elevation of Matterhorn will be the current elevation of the surveyor <em>plus </em>the height with respect to the current elevation or simply <em>h</em>. So, the elevation height is:

E=9756+h

By substitution:

\displaystyle E=9756+\frac{1500\sin(30^\circ)\sin(35^\circ)}{\sin(5^\circ)}

Now we may approximate. Use a calculator. So, the total elevation of Matterhorn is:

E=14961.78867...\approx14,962\text{ feet}

<em />

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You have 14 coins that are nickles and quarters that equals 2 dollars how do you solve using substitution?
sergejj [24]

Answer:

The question is incorrect but if half nickels and quarters exist then you could say the answer is 7.5 nickels and 6.5 quarters.

The problem:

14 coins that are either nickels are quarters

The total dollar amount of these coins is 2 dollars

Step-by-step explanation:

Let n be the number of nickels.

Let q be the number of quarters.

We are given that n+q=14 and that .05n+.25q=2.

I'm going to choose to solve the system by elimination.

Multiplying the first equation by .25 gives:

.25n+.25q=.25(14)

.25n+.25q=3.5

So let's line the equations up now:

.25n+.25q=3.5

.05n+.25q=2

---------------------If we subtract the equations, the variable q will be eliminated.                Thus, we will able to solve for the variable n.

.20n+0q=1.5

.20n=1.5

Dividing both sides by .2 gives:

n=7.5

If n=7.5 and n+q=14, then we have 7.5+q=14 by substitution property.

We can subtract 7.5 on both sides giving us:

q=14-7.5=6.5

We cannot have 7.5 nickels and 6.5 quarters but this satisfies the given information.

Check:

7.5+6.5=(7+6)+(.5+.5)=13+1=14

7.5(.05)+6.5(.25)=2

Now, the requested way which is by substitution:

Let n be the number of nickels.

Let q be the number of quarters.

We are given that n+q=14 and that .05n+.25q=2.

We can solve the first equation for either n or q and then substitute that into the other equation allowing us to solve for the other variable.

Let's solve for q by subtraction n on both sides:

q=14-n

We are going to replace q in the second equation with (14-n) giving us:

.05n+.25(14-n)=2

Distribute:

.05n+3.5-.25n=2

Combine like terms:

-0.2n+3.5=2

Subtract 3.5 on both sides:

-0.2n=-1.5

Divide both sides by -0.2:

n=7.5

Since q=14-n and n=7.5, then q=14-7.5=6.5 .

We are already check the solution above.

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