Answer :
To solve this question, let's first analyze the given equation:
[tex]\[ y = 15 \cdot 0.5^x \][/tex]
This equation models a situation where the value [tex]\( y \)[/tex] decreases exponentially as [tex]\( x \)[/tex] increases. Specifically, [tex]\( y \)[/tex] starts at 15 when [tex]\( x \)[/tex] is 0, and halves each time [tex]\( x \)[/tex] increases by 1.
Now, let's go through each situation provided to determine which one fits this model:
a. Kyle's Grandpa gave him 15 dollars for Christmas. He wanted to buy an app that costs 50 cents per month. Write an equation to model how many months will pass before all the money is gone.
- This situation involves a linear decrease in money, where [tex]\( y \)[/tex] would decrease by a constant amount (50 cents) each month. Hence, the model for this situation would be a linear equation, not an exponential equation. Thus, it does not fit the given equation.
b. The Mustangs were behind by 15 points. Write an equation to model the number of baskets they have to make to tie up the game (given they keep their opponent from scoring any points).
- This situation involves counting discrete quantities (baskets made) to reach a goal (tying the game). This would be modeled by a linear equation where each basket contributes a certain number of points towards reducing the initial deficit. This does not involve an exponential decrease, so it does not fit the given equation.
c. Valley Center had 15 feet of snow at the beginning of May. When the weather turned warm, the snow began to melt about 6 inches per day. Write the equation which models the amount of snow each day and the time it will take for the snow to melt away.
- This situation involves a linear decrease in the amount of snow, with 6 inches melting away each day. Thus, the model would be a linear equation, not an exponential equation. This does not match the given equation.
d. The Cuttie Pie Company sells their pies for \[tex]$15. Each day it doesn't sell they mark it down by 50\%. Write an equation to model the cost each passing day and when it will be given away for free. - This situation describes an exponential decrease in the price of the pie: starting at \$[/tex]15 and halving each day it doesn't sell. This matches the given equation perfectly, as the price changes by a factor of 0.5 each day ([tex]\(0.5^x\)[/tex]) starting from 15.
Therefore, the correct situation that can be modeled by the given equation [tex]\( y = 15 \cdot 0.5^x \)[/tex] is:
d. The Cuttie Pie Company sells their pies for \$15. Each day it doesn't sell they mark it down by 50\%. Write an equation to model the cost each passing day and when it will be given away for free.
[tex]\[ y = 15 \cdot 0.5^x \][/tex]
This equation models a situation where the value [tex]\( y \)[/tex] decreases exponentially as [tex]\( x \)[/tex] increases. Specifically, [tex]\( y \)[/tex] starts at 15 when [tex]\( x \)[/tex] is 0, and halves each time [tex]\( x \)[/tex] increases by 1.
Now, let's go through each situation provided to determine which one fits this model:
a. Kyle's Grandpa gave him 15 dollars for Christmas. He wanted to buy an app that costs 50 cents per month. Write an equation to model how many months will pass before all the money is gone.
- This situation involves a linear decrease in money, where [tex]\( y \)[/tex] would decrease by a constant amount (50 cents) each month. Hence, the model for this situation would be a linear equation, not an exponential equation. Thus, it does not fit the given equation.
b. The Mustangs were behind by 15 points. Write an equation to model the number of baskets they have to make to tie up the game (given they keep their opponent from scoring any points).
- This situation involves counting discrete quantities (baskets made) to reach a goal (tying the game). This would be modeled by a linear equation where each basket contributes a certain number of points towards reducing the initial deficit. This does not involve an exponential decrease, so it does not fit the given equation.
c. Valley Center had 15 feet of snow at the beginning of May. When the weather turned warm, the snow began to melt about 6 inches per day. Write the equation which models the amount of snow each day and the time it will take for the snow to melt away.
- This situation involves a linear decrease in the amount of snow, with 6 inches melting away each day. Thus, the model would be a linear equation, not an exponential equation. This does not match the given equation.
d. The Cuttie Pie Company sells their pies for \[tex]$15. Each day it doesn't sell they mark it down by 50\%. Write an equation to model the cost each passing day and when it will be given away for free. - This situation describes an exponential decrease in the price of the pie: starting at \$[/tex]15 and halving each day it doesn't sell. This matches the given equation perfectly, as the price changes by a factor of 0.5 each day ([tex]\(0.5^x\)[/tex]) starting from 15.
Therefore, the correct situation that can be modeled by the given equation [tex]\( y = 15 \cdot 0.5^x \)[/tex] is:
d. The Cuttie Pie Company sells their pies for \$15. Each day it doesn't sell they mark it down by 50\%. Write an equation to model the cost each passing day and when it will be given away for free.
