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We have been given two ratios and we are supposed to compare our ratios whether they are proportional or not.
1. We will reduce our fractions to compare our ratios. Let us simplify each of our given fractions.
Let us divide numerator and denominator with greatest common factor. We can see that 21 is GCF of our ratio.
Now we will simplify our second ratio.
GCF of our fraction is 32. Upon dividing our fraction by 32 we will get,
We can see that our both ratios are similar.
2. Now we will find decimal values of our fractions.
We can see that .
Therefore, we can say that our ratios are proportional.
<span>Given that the railroad rail is 6 kilometers before expansion.
When the railroad expands, it makes an isosceles triangle at the center of the rail.
The base of the triangle is 6 kilometers while each leg of the triangle is given by (6 + 10/100,000) / 2 = 6.0001 / 2 = 3.00005 kilometers.
[recall that 100,000 centimeters gives 1 kilometers, thus 10 centimetres gives 10/100,000 kilometers]
Taking one half of the isoceles triangle, we have a right triangle with one of the leg as 6 / 2 = 3 kilometers and the hypothenus = 3.00005 kilometers.
Thus, using the Pythagoras theorem, we find the other leg (height) of the right triangle as follows.
Let the other leg of the right triangle be h, then
Thus, the center of the rail will rise 0.01732058 kilometers above the ground.
But, there are 1,000 meters in 1 kilometers, therefore, </span><span><span>the center of the rail will rise 0.01732058 x 1,000 = 17.32 meters above the ground.</span></span>
When the railroad expands, it makes an isosceles triangle at the center of the rail.
The base of the triangle is 6 kilometers while each leg of the triangle is given by (6 + 10/100,000) / 2 = 6.0001 / 2 = 3.00005 kilometers.
[recall that 100,000 centimeters gives 1 kilometers, thus 10 centimetres gives 10/100,000 kilometers]
Taking one half of the isoceles triangle, we have a right triangle with one of the leg as 6 / 2 = 3 kilometers and the hypothenus = 3.00005 kilometers.
Thus, using the Pythagoras theorem, we find the other leg (height) of the right triangle as follows.
Let the other leg of the right triangle be h, then
Thus, the center of the rail will rise 0.01732058 kilometers above the ground.
But, there are 1,000 meters in 1 kilometers, therefore, </span><span><span>the center of the rail will rise 0.01732058 x 1,000 = 17.32 meters above the ground.</span></span>