The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 196m above the surface of Earth can see 21km to the horizon, how far can a person see to the horizon from a point 36m above the surface?
Well, let's do some math with a dash of humor to determine how far a person 36m above the surface of the Earth can see.
According to the given information, the distance that a person can see to the horizon varies directly as the square root of the height. So, we can express this relationship using a formula:
D = k√h
Where D represents the distance to the horizon, h represents the height, and k is the constant of variation.
Now, let's plug in the values we have. We know that when a person is 196m above the surface, they can see 21km (21000m) to the horizon. So we have:
21000 = k√196
To solve for k, divide both sides by √196:
21000/√196 = k
Now we can find the value of k:
k ≈ 150
Now we can use this value of k to find out how far a person 36m above the surface can see:
D = k√h
D = 150√36
D = 150 * 6
D = 900
So, a person 36m above the surface of the Earth can see approximately 900m to the horizon. Just enough to observe any rogue squirrels or runaway shopping carts heading their way!
To solve this problem, let's set up a proportion using the given information.
Let d represent the distance a person can see to the horizon from a point above the surface, and h represent the height at that point.
According to the problem, we know that the distance, d, is directly proportional to the square root of the height, h. This can be written as:
d ∝ √h
Using this proportionality, we can express it as an equation:
d = k√h
Where k is the constant of proportionality.
Now let's plug in the given values:
196 = k√(21,000)
36 = k√h
We can use these two equations to find the value of k.
Dividing the first equation by the second equation, we get:
196/36 = (√21,000)/(√h)
Simplifying:
5.44 ≈ (√21,000)/(√h)
Now we can solve for h by cross multiplying:
5.44√h = √21,000
Square both sides to eliminate the square root:
29.6h = 21,000
Divide both sides by 29.6:
h ≈ 709.46
Now that we have the value of h, we can plug it back into the equation to find the distance, d, a person can see to the horizon from a point 36m above the surface:
d = k√h
d = k√709.46
Let's substitute the value of d from the first equation:
196 = k√21,000
Now we can solve for k:
k = 196/√21,000
Using a calculator, approximate this value:
k ≈ 0.0816
Finally, we can find the distance, d, a person can see to the horizon from a point 36m above the surface:
d = k√h
d = 0.0816√36
Calculating:
d ≈ 0.0816 * 6
d ≈ 0.4896
Therefore, a person 36m above the surface of the Earth can see approximately 0.4896 km to the horizon.
To solve this problem, we can use the concept of direct variation. Direct variation means that two variables are connected in such a way that when one variable increases (or decreases) by a certain factor, the other variable also increases (or decreases) by the same factor.
In this case, we are given that the distance someone can see to the horizon (let's call it D) varies directly as the square root of the height (let's call it H). Mathematically, we can express this as:
D = k * sqrt(H),
where k is the constant of variation.
Now, we are given that when a person is 196m above the surface of Earth, they can see 21km to the horizon. Let's use this information to find the constant of variation, k.
21km = k * sqrt(196m),
To find k, we can solve this equation:
sqrt(196m) = 21km / k.
Squaring both sides of the equation, we get:
196m = (21km / k)^2.
Now we can solve for k:
k = 21km / sqrt(196m).
Now that we have the value of k, we can use it to find the distance a person can see to the horizon when they are 36m above the surface:
D = k * sqrt(H),
D = (21km / sqrt(196m)) * sqrt(36m),
Simplifying,
D = (21km / sqrt(196m)) * 6m,
D = (21km / sqrt(4^2 * 49)) * 6m,
D = (21km / (2 * 7)) * 6m,
D = (21km / 14) * 6m,
D = 3km * 6m,
D = 18km.
Therefore, a person 36m above the surface of Earth can see 18km to the horizon.
d = k√h, so d/√h is constant
You want d such that
d/√36 = 21/√196