Answered

2. Two math students were asked to write an exponential growth equation that had a starting value of 300 and a growth rate of [tex]$2 \%$[/tex]. Pierre thinks the answer is [tex]$y=300(1.02)^x$[/tex] and Scott thinks that the answer is [tex][tex]$y=300(1.2)^x$[/tex][/tex]. Are either of them right, and why?

A. Neither are right; [tex]2 \%[/tex] written as a decimal is 0.02, so the base should just be 0.02.
B. Scott is right; [tex]2 \%[/tex] written as a decimal is 0.2, so the base should be [tex]$1+0.2$[/tex].
C. Pierre is right; [tex]2 \%[/tex] written as a decimal is 0.02, so the base should be [tex]$1+0.02$[/tex].
D. Neither are right; [tex]2 \%[/tex] written as a decimal is 02, so the base should just be 02.



Answer :

GinnyAnswer
Let's go through the problem step-by-step to determine if Plerre or Scott have the correct exponential growth equation for a starting value of 300 and a growth rate of 2%.

1. Identify the given values:
- Starting value [tex]\( y_0 \)[/tex] = 300
- Growth rate in percent [tex]\( g \)[/tex] = 2%

2. Convert the growth rate to a decimal:
- The growth rate as a percent (2%) can be converted to a decimal by dividing by 100:
[tex]\[ g_{\text{decimal}} = \frac{2}{100} = 0.02 \][/tex]

3. Understand the general form of the exponential growth equation:
- The general form of the exponential growth equation is:
[tex]\[ y = y_0 \cdot (1 + g_{\text{decimal}})^x \][/tex]
- Substituting the given values:
[tex]\[ y = 300 \cdot (1 + 0.02)^x \][/tex]

4. Check Plerre's equation:
- Plerre's equation: [tex]\( y = 300(1.02)^x \)[/tex]
- Compare this with the correct form [tex]\( y = 300 \cdot (1 + 0.02)^x \)[/tex]:
[tex]\[ y = 300 \cdot (1.02)^x \][/tex]
- Plerre's equation matches the correct exponential growth equation.

5. Check Scott's equation:
- Scott's equation: [tex]\( y = 300(1.2)^x \)[/tex]
- Compare this with the correct form [tex]\( y = 300 \cdot (1 + 0.02)^x \)[/tex]:
[tex]\[ y = 300 \cdot (1.2)^x \][/tex]
- Scott's equation does not match the correct exponential growth equation, as it uses a base of 1.2 instead of 1.02, indicating an incorrect growth rate of 20% instead of 2%.

6. Conclusion:
- Plerre is correct because his equation [tex]\( y = 300(1.02)^x \)[/tex] uses the correct base of [tex]\( 1 + 0.02 \)[/tex].
- Scott is incorrect because his equation [tex]\( y = 300(1.2)^x \)[/tex] uses an incorrect base of 1.2, which corresponds to a 20% growth rate.

Therefore, the correct statement is:
- Plerre is right. [tex]$2\%$[/tex] written as a decimal is 0.02, so the base should be [tex]\(1 + 0.02\)[/tex].