Question
Karen is a product tester at Lulu's Lotions & More. She's testing a new line of bath fizzers and needs to measure how quickly they dissolve in water. The first bath fizzer she tested weighed 5 ounces to start. After one minute in water, it decreased by half, and Karen expects this rate of decrease to continue.
Write an exponential equation in the form y=a(b)x that can model the weight of the bath fizzer, y, after x minutes in water.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
How much will the bath fizzer weigh after 3 minutes in water?
ounces
Write an exponential equation in the form y=a(b)x that can model the weight of the bath fizzer, y, after x minutes in water.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
How much will the bath fizzer weigh after 3 minutes in water?
ounces
Answers
GPT-4o mini
To model the weight of the bath fizzer over time with an exponential equation, we start with the initial weight of the fizzer and the information about how it dissolves.
The initial weight, \(a\), is 5 ounces. The problem states that after one minute, the weight decreases by half. Therefore, the multiplicative factor \(b\) is 0.5 since the weight is halved.
The exponential equation can be written in the form:
\[
y = a(b)^x
\]
Substituting the known values into the equation:
\[
y = 5(0.5)^x
\]
Now, to find the weight of the bath fizzer after 3 minutes in water, we will substitute \(x = 3\) into the equation:
\[
y = 5(0.5)^3
\]
Calculating \( (0.5)^3 \):
\[
(0.5)^3 = 0.125
\]
Now substitute back into the equation:
\[
y = 5 \times 0.125 = 0.625
\]
Thus, the weight of the bath fizzer after 3 minutes in water is:
\[
\text{Weight after 3 minutes} = 0.625 \text{ ounces}
\]
To summarize:
- The equation is \(y = 5(0.5)^x\).
- The weight of the bath fizzer after 3 minutes is \(0.625\) ounces.
The initial weight, \(a\), is 5 ounces. The problem states that after one minute, the weight decreases by half. Therefore, the multiplicative factor \(b\) is 0.5 since the weight is halved.
The exponential equation can be written in the form:
\[
y = a(b)^x
\]
Substituting the known values into the equation:
\[
y = 5(0.5)^x
\]
Now, to find the weight of the bath fizzer after 3 minutes in water, we will substitute \(x = 3\) into the equation:
\[
y = 5(0.5)^3
\]
Calculating \( (0.5)^3 \):
\[
(0.5)^3 = 0.125
\]
Now substitute back into the equation:
\[
y = 5 \times 0.125 = 0.625
\]
Thus, the weight of the bath fizzer after 3 minutes in water is:
\[
\text{Weight after 3 minutes} = 0.625 \text{ ounces}
\]
To summarize:
- The equation is \(y = 5(0.5)^x\).
- The weight of the bath fizzer after 3 minutes is \(0.625\) ounces.