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Answer :
To solve the problem, we'll start by understanding the function given: [tex]\( f(t) = 81(0.83)^t \)[/tex]. This function represents how the temperature of the ice cream, below room temperature, changes over time [tex]\( t \)[/tex].
Let's analyze the required points step-by-step:
1. Figure out where the graph crosses the [tex]\( y \)[/tex]-axis:
- The graph of the function [tex]\( f(t) \)[/tex] crosses the [tex]\( y \)[/tex]-axis when [tex]\( t = 0 \)[/tex]. To find this point, we substitute [tex]\( t = 0 \)[/tex] into the function.
- [tex]\( f(0) = 81(0.83)^0 \)[/tex]. Since any number raised to the power of 0 is 1, this simplifies to:
- [tex]\( f(0) = 81 \times 1 = 81 \)[/tex].
2. Interpret the meaning of this point in the context of the problem:
- The calculated value [tex]\( f(0) = 81 \)[/tex] tells us that at time [tex]\( t = 0 \)[/tex], the below-room temperature of the ice cream is [tex]\( 81^{\circ}C \)[/tex].
Therefore, the graph of [tex]\( f(t) \)[/tex] crosses the [tex]\( y \)[/tex]-axis at [tex]\( (0, 81) \)[/tex], meaning that at time [tex]\( t = 0 \)[/tex], the ice cream is [tex]\( 81^{\circ}C \)[/tex] below room temperature.
Hence, the correct interpretation in the context of the problem is:
- [tex]\( f(0) = 81 \)[/tex]; the below-room temperature of the ice cream at time [tex]\( t = 0 \)[/tex].
Let's analyze the required points step-by-step:
1. Figure out where the graph crosses the [tex]\( y \)[/tex]-axis:
- The graph of the function [tex]\( f(t) \)[/tex] crosses the [tex]\( y \)[/tex]-axis when [tex]\( t = 0 \)[/tex]. To find this point, we substitute [tex]\( t = 0 \)[/tex] into the function.
- [tex]\( f(0) = 81(0.83)^0 \)[/tex]. Since any number raised to the power of 0 is 1, this simplifies to:
- [tex]\( f(0) = 81 \times 1 = 81 \)[/tex].
2. Interpret the meaning of this point in the context of the problem:
- The calculated value [tex]\( f(0) = 81 \)[/tex] tells us that at time [tex]\( t = 0 \)[/tex], the below-room temperature of the ice cream is [tex]\( 81^{\circ}C \)[/tex].
Therefore, the graph of [tex]\( f(t) \)[/tex] crosses the [tex]\( y \)[/tex]-axis at [tex]\( (0, 81) \)[/tex], meaning that at time [tex]\( t = 0 \)[/tex], the ice cream is [tex]\( 81^{\circ}C \)[/tex] below room temperature.
Hence, the correct interpretation in the context of the problem is:
- [tex]\( f(0) = 81 \)[/tex]; the below-room temperature of the ice cream at time [tex]\( t = 0 \)[/tex].
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