Quiz 0323

If a certain rock weighs 12 lb and has maximum length 9 inches, then what would we expect to be the weight of a geometrically similar rock whose maximum length is 27 inches?

The weight of a rock resides in the little cubes of which it is composed.  So the proportionality here is

• y = k x^3.

We substitute the known information to get the equation

• 12 lb = k * (9 inches)^3,

and solve for k, obtaining

• k = 12 lb / (9 inches)^3 = 12 lb / (729 in^3) = .0166 lb / in^3, approx..

So our equation is now

• y = .0166 lb / in^3 * x^3, or to give it more meaning
• weight = .0166 lb / in^3 * length^3.

If length is 27 in we therefore get

• weight = .0166 in^3 * (27 in)^3 = ... = 324 lb.

If it takes $1.11 worth of suntan lotion to fully protect the first rock from the sun for a week, then under identical conditions how much will it cost to protect the second rock for a week?

Suntan lotion, as we know, goes on the surface of the rock.  You CAIN'T fill a rock with suntan lotion.

The surface of the rock is covered by tiny squares.  So we have a proportionality y = k x^2, or to give it more meaning cost = k * length^2.

Substituting our given information we have $1.11 = k * (9 in)^2.  Solving for k we get k = $1.11 / (9 in)^2 = $.0137 / in^2.  So our proportionality is

• cost = $.0137 / in^2 * length^2.

At a length of 27 inches we therefore have

• cost = $.0137 / in^2 * (27 in)^2 = $9.99.

Solve each of the following equations:

• x ^ 5 / 8 = 12

Note that x^5/8 = 12 means (x^5) / 8 = 12, since exponentiation precedes division.

If we wanted the exponent to be 5/8 we would have to write this x^(5/8) = 12.

We solve this equation as indicated in the figure below.

• 12 y^(3/7) = 9.

• (8 x) ^ -3 = 11

Raise both sides to the -(1/3), obtaining

8x = 11 ^-(1/3).

Divide both sides by 8 to get

x = 11 ^-(1/3) / 8 = .0562 approx.

• 14 x ^ -.2 = 77

Divide both sides by 14 to get

x^-.2 = 77 / 14. 

Raise both sides to the 1 / (-.2) to get

x = (77/14)^(1 / (-.2)) = .0001987.

Note that in order to fully illustrate the steps we haven't done simplifications such as 77/14 = 11/2, or 1 / (-.2) = -5. 

• 4 x ^ (5/3) = 3 

Divide both sides by 4 to get

x^(5/3) = 3/4.

Raise both sides to the 3/5 power to get

x = (3/4)^(3/5) = .8415 approx..

• 4 ( 6 x / 7) ^ (-5/2) = 2

Divide both sides by 4 to get

(6x / 7)^(-5/2) = 2/4. 

Raise both sides to the -2/5 power to get

6x / 7 = (2/4)^(-2/5).

Multiply both sides by 7/6 to get

x = (2/4)^(-2/5) * 7/6 = 1.540.

A man 6 feet tall weighs 170 lbs and has about 20 square feet of skin surface.   If that man was scaled up to a height of 7 feet how much would he weigh and how many square feet of skin surface will he have?  How many times as much sweat would you expect the big guy to produce?

People are made up of little cubes so the proportionality is weight = k * height^3. 

Substituting we have 170 lbs = k * (6 feet)^3, which gives us k = 170 lbs / (6 ft)^3 = .8 approx..

• So our proportionality is weight = .8 * height^3.

Height 7 ft gives us

• weight = .8 * (7 ft)^3 = 270 lbs approx..

The big guy will produce 270 / 170 times as much body heat so in order to stay cool he will need to evaporate twice as much water, which would imply twice as much sweat.  We could have obtained the same ratio (up to the accuracy of our approximations) by calculating (7/6)^3.

Note however that he doesn't have twice as much surface area, since surface area is proportional to the square of linear dimensions.  So there will be (7/6)^3 times as much sweat distributed over (7/6)^2 times as much area, giving us (7/6)^3 / (7/6)^2 = 7/6 times as much sweat density on the big guy's skin.