class 051024
If y is proportional to x^(-2) and y = 12 when x = 30, then
If y is proportional to x^(-2) then
- y = k x^(-2) for some constant value of k.
Plugging in y = 12 and x = 30 this gives us
- 12 = k * 30^(-2). We can divide by 30^(-2) to get
- k = 12 / (30^(-2)). Calculator will do this but to
illustrate basic arithmetic involved here:
- 12 / (30^(-2)) = 12 / (1 / 30^2) = 12 * 30^2 / 1 = 12 *
30^2 = 12 * 900 = 10,800.
- So k = 10,800.
-
- Substituting this into the original form we get
- y = 10,800 * x^(-2)
- what is the value of y when x = 90?
- Plugging in x = 90 to the form y = 10,800 x^(-2) we get y
= 1.33..
The mass of a 4-inch sphere of concrete is 1.7 kg.
- What is the proportionality between the mass of a sphere and its radius?
- Mass occupies tiny cubes, so mass = k * radius^3.
- What is the value of the proportionality constant for this situation?
- We don't say what the 4 inches is. Assume it's
diameter, so the radius is 2 inches.
- When radius is 2 inches the mass is 1.7 kg.
- So we have
- 1.7 = k * 2^3 or
- k = 1.7 / 2^3 = 1.7 / 8 = .21.
- The proportionality equation becomes mass = .21 *
radius^3.
- What is the mass of a 3-inch sphere of concrete?
- The 3-inch sphere has radius 1.5 inches so its mass is
- mass = .21 * 1.5^3 = .21 * 3.375 = .7 kg, approx..
It takes 1200 small tiles to tile a certain bathtub, which is 6 feet long.
How many identical tiles will it take to tile a geometrically similar bathtub
which is 8 feet long?
We're scaling up little squares. The scale increases by
a factor of 8/6 = 4/3.
Tiles cover area, and area is related to length dimension by
area = k * length^2.
If area is number of tiles, then we have
number of tiles = k * length^2 so
1200 = k * 6^2 and
k = 1200 / 6^2 = 33.3...
Thus when length = 8 ft we have
# of tiles = 33.3 * length^2 = 33.3 * 8^2 = 2133.