The amount of oil used by a ship traveling at a uniform speed varies jointly with the distance and the square of the speed. If the ship uses 450 barrels of oil in traveling 500 miles at 44mph, determine how many barrels of oil are used when the ship travels 440 miles at 22mph
B = barrels = k d v^2
450 = k (500 * 44 *44)
so
k = 450 / (500 * 44 *44)
then
B = k (440 * 22 * 22)
B = [ 450 / (500 * 44 *44) ] (440 * 22 * 22)
B = 450 (44/50)(1/4)
B = 450 (11/50)
B = 99
or just say half the speed is 1/4 the fuel rate per mile
so right off
B = 450 (440/500)(1/4) = 450 (110/500) = 99
15.2 barrels
To solve this problem, we can use the concept of joint variation, which states that a variable is directly proportional to the product of two or more other variables.
Let's define the following variables:
- Let "B" represent the number of barrels of oil used.
- Let "D" represent the distance traveled.
- Let "S" represent the speed of the ship.
As stated in the problem, the amount of oil used jointly varies with the distance and the square of the speed. Mathematically, this can be represented as:
B = kDS^2
where "k" is a constant of variation.
Now, let's use the given information to calculate the value of "k".
When the ship travels 500 miles at 44 mph, it uses 450 barrels of oil. Plugging these values into the equation:
450 = k * 500 * 44^2
450 = k * 500 * 1936
450 = k * 968,000
Now, we can solve for the value of "k":
k = 450 / 968,000
k ≈ 0.00046488
So, the equation becomes:
B = 0.00046488 * D * S^2
Now, let's use this equation to determine how many barrels of oil are used when the ship travels 440 miles at 22 mph.
Plugging these values into the equation:
B = 0.00046488 * 440 * 22^2
B = 0.00046488 * 440 * 484
B ≈ 48.262
Therefore, when the ship travels 440 miles at 22 mph, it will use approximately 48.262 barrels of oil.