The price of Stock A at 9 AM was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at... The price of Stock A at 9 AM was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at the rate of $0.14 per hour. If the stocks continue to increase and decrease at the same rates, in how many hours will the prices of the stocks be the same?
Understand the Problem
The question is asking for the number of hours it will take for Stock A and Stock B to have the same price, given their rates of increase and decrease over time.
Answer
The prices of the stocks will be the same in approximately $2.85$ hours.
Answer for screen readers
The prices of the stocks will be the same in approximately $2.85$ hours.
Steps to Solve
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Determine Initial Prices and Rates The price of Stock A at 9 AM is $12.73, and it increases by $0.06 per hour. The price of Stock B at 12 PM is $13.48, and it decreases by $0.14 per hour.
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Establish the Time Difference From 9 AM to 12 PM, there are 3 hours. Therefore, by 12 PM, the price of Stock A will have increased by: $$ \text{Increase for Stock A} = 0.06 \times 3 = 0.18 $$
So, the price of Stock A at 12 PM is: $$ \text{Price of Stock A at 12 PM} = 12.73 + 0.18 = 12.91 $$
- Write Equations for Price Changes Let $t$ be the number of hours after 12 PM.
The price of Stock A after $t$ hours will be: $$ \text{Price of Stock A} = 12.91 + 0.06t $$
The price of Stock B after $t$ hours will be: $$ \text{Price of Stock B} = 13.48 - 0.14t $$
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Set the Prices Equal to Each Other To find when the prices are the same, set the equations equal: $$ 12.91 + 0.06t = 13.48 - 0.14t $$
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Solve for $t$ Combine like terms: $$ 0.06t + 0.14t = 13.48 - 12.91 $$ $$ 0.20t = 0.57 $$ Now, divide both sides by 0.20: $$ t = \frac{0.57}{0.20} = 2.85 $$
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Final Calculation The stocks will be at the same price in approximately 2.85 hours after 12 PM, which translates to about 2 hours and 51 minutes.
The prices of the stocks will be the same in approximately $2.85$ hours.
More Information
The rates of increase and decrease for stocks can lead to interesting dynamics in pricing. Understanding how to set up equations and solve for variables is crucial in financial mathematics.
Tips
- Confusing the rates of change with the initial prices.
- Neglecting the time difference when calculating price adjustments.
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