Type the correct answer in each box. Use numerals instead of words. Two chain restaurant brands, Joey's Refuge and Kane's Deli, were founded in the same year, although some outlets of Kane's Deli were operating under a different name before Kane's Deli was founded. The owner of Joey's Refuge uses the equation below to represent the number of restaurants, $y$, he owns $x$ years after the brand's founding. $y=2^x$ The owner of Kane's Deli uses the equation below to represent the number of restaurants, $y$, he owns $x$ years after the original brand's founding. $y=4 x+12$ Use graphing technology, and complete the given statements. After $\square$ years, Joey's Refuge and Kane's Deli will own the same number of restaurants. At that time, each chain will own $\square$ restaurants.
Answer (2)
Set up the equations for Joey's Refuge ( y = 2 x ) and Kane's Deli ( y = 4 x + 12 ).
Equate the two expressions to find when they have the same number of restaurants: 2 x = 4 x + 12 .
Solve for x by testing integer values: x = 5 .
Determine the number of restaurants at that time: y = 32 . The final answer is 5 and 32 .
Explanation
Problem Analysis Let's analyze the problem. We have two equations representing the number of restaurants for two different chains, Joey's Refuge and Kane's Deli, y = 2 x and y = 4 x + 12 , respectively, where x is the number of years after the brands' founding. We need to find the year when both chains have the same number of restaurants. This means we need to solve the system of equations to find the value of x and y where the two equations are equal.
Setting up the Equation To find the point where the number of restaurants is the same for both chains, we need to solve the equation 2 x = 4 x + 12 . We can solve this by trying different values of x or using a numerical method. We can start by testing integer values of x .
Solving for x Let's test some values of x :
If x = 1 , then 2 1 = 2 and 4 ( 1 ) + 12 = 16 . Not equal. If x = 2 , then 2 2 = 4 and 4 ( 2 ) + 12 = 20 . Not equal. If x = 3 , then 2 3 = 8 and 4 ( 3 ) + 12 = 24 . Not equal. If x = 4 , then 2 4 = 16 and 4 ( 4 ) + 12 = 28 . Not equal. If x = 5 , then 2 5 = 32 and 4 ( 5 ) + 12 = 32 . Equal!
Finding the Solution We found that when x = 5 , both equations give y = 32 . Therefore, after 5 years, both Joey's Refuge and Kane's Deli will have the same number of restaurants, which is 32.
Examples
Imagine you are deciding between two investment opportunities. One investment grows exponentially, similar to Joey's Refuge, while the other grows linearly, like Kane's Deli. Initially, the linear investment might seem better due to a higher starting value. However, understanding when the exponential investment surpasses the linear one is crucial for making informed long-term decisions. This problem demonstrates how mathematical models can help predict future outcomes and guide strategic choices in business and finance.
Joey's Refuge and Kane's Deli will have the same number of restaurants after 5 years. At that time, both chains will own 32 restaurants.
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