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Answer :
Certainly! Let's analyze the given equation [tex]\( y = 20000(0.95)^z \)[/tex] to determine whether it represents growth or decay.
1. Understand the equation structure:
- The equation is in the form of an exponential function, which typically looks like [tex]\( y = a \cdot b^z \)[/tex], where [tex]\( a \)[/tex] is the initial amount, [tex]\( b \)[/tex] is the base or multiplication factor, and [tex]\( z \)[/tex] is the exponent.
2. Identify the initial amount:
- Here, [tex]\( a = 20000 \)[/tex]. This is the starting value before any changes are applied over time.
3. Analyze the base [tex]\( b \)[/tex]:
- The base of the exponent in this equation is [tex]\( 0.95 \)[/tex].
4. Determine growth or decay:
- In exponential functions, if the base [tex]\( b \)[/tex] is greater than 1, the function represents growth, meaning the quantity increases over time.
- If the base [tex]\( b \)[/tex] is less than 1, the function represents decay, meaning the quantity decreases over time.
5. Conclusion:
- Since [tex]\( 0.95 \)[/tex] is less than 1, this indicates decay. As you apply higher values of [tex]\( z \)[/tex], the overall value of [tex]\( y \)[/tex] decreases.
Therefore, the equation [tex]\( y = 20000(0.95)^z \)[/tex] represents decay.
1. Understand the equation structure:
- The equation is in the form of an exponential function, which typically looks like [tex]\( y = a \cdot b^z \)[/tex], where [tex]\( a \)[/tex] is the initial amount, [tex]\( b \)[/tex] is the base or multiplication factor, and [tex]\( z \)[/tex] is the exponent.
2. Identify the initial amount:
- Here, [tex]\( a = 20000 \)[/tex]. This is the starting value before any changes are applied over time.
3. Analyze the base [tex]\( b \)[/tex]:
- The base of the exponent in this equation is [tex]\( 0.95 \)[/tex].
4. Determine growth or decay:
- In exponential functions, if the base [tex]\( b \)[/tex] is greater than 1, the function represents growth, meaning the quantity increases over time.
- If the base [tex]\( b \)[/tex] is less than 1, the function represents decay, meaning the quantity decreases over time.
5. Conclusion:
- Since [tex]\( 0.95 \)[/tex] is less than 1, this indicates decay. As you apply higher values of [tex]\( z \)[/tex], the overall value of [tex]\( y \)[/tex] decreases.
Therefore, the equation [tex]\( y = 20000(0.95)^z \)[/tex] represents decay.
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