dancinggeometry
The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of 5x cents per day while the price of wheat is decreasing at a constant rate of \(x\sqrt{2} - x\) cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?
(A) $4.50
(B) $5.10
(C) $5.30
(D) $5.50
(E) $5.60
Distance between the two prices = 580 - 320 = 260 cents.
We can treat this as a RATE problem, with the prices of corn and wheat TRAVELING TOWARD EACH OTHER to meet somewhere between 320 cents and 580 cents.
Let x=1.
Rate of increase for corn = 5x = 5*1 = 5 cents per day.
Rate of decrease for wheat = \(x\sqrt{2} - x\) ≈ 1.4x - x = 0.4x = 0.4(1) = 0.4 cents per day
For the two prices to meet, they must WORK TOGETHER to cover the 260-cent distance between them.
When elements work together, ADD THEIR RATES:
Combined rate for corn and wheat = 5 + 0.4 = 5.4 cents per day.
Of every 5.4 cents covered by the two prices as they work together to meet, wheat covers 0.4 cents.
Implication:
Wheat will cover \(\frac{0.4}{5.4} = \frac{2}{27}\) of the 260-cent distance between corn and wheat.
Thus:
Distance covered by wheat = \(\frac{2}{27} * 260 ≈ 2 * \frac{261}{27} = 2 * \frac{29}{3} ≈ 2*10 = 20\) cents
Since the 580-cent wheat decreases by about 20 cents to meet the price of corn, we get:
580 - 20 = 560 cents