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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 \(5\mathrm{x}\) cents per day while the price of wheat is decreasing at a constant rate of \((\mathrm{x}\sqrt{2} - \mathrm{x})\) cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?
Let's understand what's happening in everyday terms: We have two different grains with different prices that are changing over time. Think of it like two cars driving toward each other on a road - one starting from a lower position moving up, and one starting from a higher position moving down. We want to find where they meet.
Currently:
The prices change daily:
We need to find the price when these two meet - when a bushel of corn costs exactly the same as a peck of wheat.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
Now let's track how each price changes day by day. If we call the number of days 't', then:
Corn's price after t days:
Starts at \(\$3.20\) and increases by \(5x\) cents per day
Price = \(\$3.20 + (5x \times t)\) cents
Converting cents to dollars: Price = \(\$3.20 + \$0.05xt\)
Wheat's price after t days:
Starts at \(\$5.80\) and decreases by \((x\sqrt{2} - x)\) cents per day
Price = \(\$5.80 - (x\sqrt{2} - x) \times t\) cents
Converting cents to dollars: Price = \(\$5.80 - \$0.01(x\sqrt{2} - x)t\)
The prices are equal when corn's price equals wheat's price. Setting our equations equal:
\(\$3.20 + \$0.05xt = \$5.80 - \$0.01(x\sqrt{2} - x)t\)
Let's solve for t step by step:
\(\$0.05xt + \$0.01(x\sqrt{2} - x)t = \$5.80 - \$3.20\)
\(\$0.05xt + \$0.01(x\sqrt{2} - x)t = \$2.60\)
Factor out t:
\(t[\$0.05x + \$0.01(x\sqrt{2} - x)] = \$2.60\)
\(t[\$0.05x + \$0.01x\sqrt{2} - \$0.01x] = \$2.60\)
\(t[\$0.04x + \$0.01x\sqrt{2}] = \$2.60\)
\(t \times x[\$0.04 + \$0.01\sqrt{2}] = \$2.60\)
Since \(\sqrt{2} \approx 1.414\):
\(t \times x[\$0.04 + \$0.01(1.414)] = \$2.60\)
\(t \times x[\$0.04 + \$0.01414] = \$2.60\)
\(t \times x[\$0.05414] = \$2.60\)
Therefore: \(t \times x = \$2.60 \div \$0.05414 \approx 48.02\)
Now we substitute back into either price equation. Using the corn equation is simpler:
Common price = \(\$3.20 + \$0.05 \times (t \times x)\)
Common price = \(\$3.20 + \$0.05 \times 48.02\)
Common price = \(\$3.20 + \$2.40\)
Common price = \(\$5.60\)
Let's verify with the wheat equation:
We need \((x\sqrt{2} - x) \times t = (1.414x - x) \times t = 0.414x \times t = 0.414 \times 48.02 \approx 19.88\)
Wheat price = \(\$5.80 - \$0.01 \times 19.88 = \$5.80 - \$0.20 = \$5.60\) ✓
The approximate price when a bushel of corn costs the same amount as a peck of wheat is \(\$5.60\).
This matches answer choice (E) \(\$5.60\).
1. Misinterpreting the direction of price changes: Students may confuse which price is increasing and which is decreasing. The problem states corn price is "increasing" at \(5x\) cents per day and wheat price is "decreasing" at \((x\sqrt{2} - x)\) cents per day. Some students might accidentally reverse these directions when setting up their equations, leading to corn decreasing and wheat increasing.
2. Confusion about what the question is asking for: The question asks for "the approximate price when a bushel of corn costs the same amount as a peck of wheat." Students might misinterpret this as asking for the number of days until the prices are equal, or they might think they need to find both individual prices rather than the common price at the meeting point.
3. Unit conversion oversight: Students may forget that the rate changes are given in cents per day while the initial prices are in dollars. This could lead them to set up equations mixing cents and dollars without proper conversion, resulting in equations like \(\$3.20 + 5xt\) instead of \(\$3.20 + \$0.05xt\).
1. Algebraic manipulation errors when combining terms: When simplifying the equation \(t[\$0.05x + \$0.01(x\sqrt{2} - x)] = \$2.60\), students often make mistakes in distributing and combining like terms. They might incorrectly calculate \(\$0.05x + \$0.01x\sqrt{2} - \$0.01x\), forgetting to subtract the \(\$0.01x\) term or making sign errors.
2. Approximation errors with \(\sqrt{2}\): Students may use an incorrect approximation for \(\sqrt{2}\) (such as 1.4 instead of 1.414) or make calculation errors when computing \(\$0.01\sqrt{2}\). This leads to an incorrect coefficient in the final equation and subsequently wrong values for the product tx.
3. Arithmetic errors in the final calculation: Even with the correct setup, students may make computational mistakes when calculating \(tx \approx 48.02\) from the division \(\$2.60 \div \$0.05414\), or when finding the final price \(\$3.20 + \$0.05 \times 48.02 = \$5.60\).
1. Selecting based on intermediate calculations: Students might accidentally select an answer choice that matches one of their intermediate values, such as the initial price difference (\(\$2.60\)), the value of tx (\(\approx 48\)), or the price increase amount (\(\$2.40\)), rather than the final common price of \(\$5.60\).
2. Rounding confusion: Since the question asks for an "approximate" price and the calculated answer is exactly \(\$5.60\), students might second-guess themselves and choose a nearby value like \(\$5.50\) (choice D) thinking that their exact calculation must be slightly off and needs rounding.