An aeroplane is flying at a height
of 300 metres above ground. Flying at this height, the angles
of depression from the aeroplane to the nearest points on opposite banks of the
river are 45 degrees and 60 degrees, respectively. Find the width of the river. Use the square root of three is
equal to 1.732.
We can start with a sketch. We have an aeroplane that is flying
at a height of 300 metres. We can imagine the flight path of
the plane being parallel to the water below — this is what a 45-degree angle of
depression would look like — to the nearest point on the bank of the river. We’ll call it 𝐸. This angle also measures 45 degrees
because these two angles are alternate interior angles.
The same is true on the other
side. We have an angle of depression of
60 degrees to the point on the bank, which we’ll call 𝑊. And we’ll label its alternate
interior angle as 60 degrees. What represents the width of the
river in our diagram?
The width of the river is the
distance from one side to the other, from our 𝑊 to our 𝐸. We can use right triangles to help
us solve this problem. The height of the aeroplane is a
perpendicular distance to the river. That helps us to see that we’re
dealing with two right triangles. We’ll label one of the lengths
lowercase 𝑒 and the other side lowercase 𝑤.
Starting on the left, we have a
right triangle. From our angle, we know the
opposite side length. And we want to know the adjacent
side length. This ratio, opposite and adjacent,
tells us that we’ll be working with a tangent ratio. Tangent of 60 degrees equals 300
over our unknown value. We know tangent of 60 degrees
equals the square root of three. We’ll substitute the square root of
three for the tangent of 60 degrees in our problem. Our new equation says the square
root of three equals 300 over 𝑤. I’ll multiply the right side by
𝑤. And the left side, on the right,
the 𝑤s cancel out. And our new statement says the
square root of three times 𝑤 equals 300.
To get 𝑤 by itself, we divide by
the square root of three on the left and the right. And we find out that 𝑤 equals 300
over the square root of three. We don’t really wanna keep the
square root of three in the denominator. So we rationalise by multiplying
the square root of three over the square root of three. Our numerator becomes 300 times the
square root of three. And in the denominator, the square
root of three times the square root of three equals three.
We recognise that 300 is divisible
by three. 300 divided by three is 100. Our simplified side length is then
100 times the square root of three. We plug in 100 times the square
root of three into our diagram.
Looking at the other side of the
triangle, we recognise something. This right triangle is a 45-45-90
triangle. And we know that, in a 45-45-90
triangle, the ratio of side lengths are always one to one to the square root of
two. The two smaller sides of a
45-45-90-degree triangle are equal to each other. And that means side 𝑒 is 300
To find the width of the river, we
need to combine both of these sides: 100 times the square root of three plus
300. To solve this problem, we can
factor out 100: 100 times the square root of three plus three. We’ve been given an approximation
for the square root of three. Substituting that in, we have 1.732
plus three. 1.732 plus three equals 4.732. If we multiply 100 times 4.732, we
move the decimal two places to the right and we find that the width of the river is