Question Exponential drug decay: A patient has $400 \mathrm{mg}$ of a drug decreasing by $7 \%$ per hour. Identify the correct equation: $f(x)=400 \cdot 0.93^{x}$

Studdy Solution

STEP 1

Assumptions
1. The initial amount of the drug in the patient's system is $400 \mathrm{mg}$ .
2. The drug decreases by $7\%$ each hour.
3. The function $f(x)$ represents the amount of drug in the patient's system after $x$ hours.

STEP 2

Understand the nature of exponential decay. When a quantity decreases by a certain percentage each time period, it can be modeled by an exponential decay function.
$f(x) = P \cdot (1 - r)^x$
where: - $P$ is the initial amount, - $r$ is the decay rate (as a decimal), - $x$ is the number of time periods.

STEP 3

Convert the percentage decrease to a decimal to find the decay rate.
$7\% = \frac{7}{100} = 0.07$

STEP 4

Subtract the decay rate from 1 to find the base of the exponential function.
$1 - r = 1 - 0.07 = 0.93$

STEP 5

Write the exponential decay function using the initial amount and the base found in the previous steps.
$f(x) = 400 \cdot 0.93^x$

STEP 6

Identify the correct equation from the given options that matches the exponential decay function derived in the previous step.
The correct equation is:
$f(x) = 400 \cdot 0.93^x$

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Question Exponential drug decay: A patient has $400 \mathrm{mg}$ of a drug decreasing by $7 \%$ per hour. Identify the correct equation: $f(x)=400 \cdot 0.93^{x}$

Studdy Solution

STEP 1

Assumptions
1. The initial amount of the drug in the patient's system is $400 \mathrm{mg}$ .
2. The drug decreases by $7\%$ each hour.
3. The function $f(x)$ represents the amount of drug in the patient's system after $x$ hours.

STEP 2

STEP 3

Convert the percentage decrease to a decimal to find the decay rate.
$7\% = \frac{7}{100} = 0.07$

STEP 4

Subtract the decay rate from 1 to find the base of the exponential function.
$1 - r = 1 - 0.07 = 0.93$

STEP 5

Write the exponential decay function using the initial amount and the base found in the previous steps.
$f(x) = 400 \cdot 0.93^x$

STEP 6

Identify the correct equation from the given options that matches the exponential decay function derived in the previous step.
The correct equation is:
$f(x) = 400 \cdot 0.93^x$

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Question Exponential drug decay: A patient has 400mg400 \mathrm{mg}400mg of a drug decreasing by 7%7 \%7% per hour. Identify the correct equation: f(x)=400⋅0.93xf(x)=400 \cdot 0.93^{x}f(x)=400⋅0.93x

Studdy Solution

STEP 1

Assumptions1. The initial amount of the drug in the patient's system is 400mg400 \mathrm{mg}400mg.2. The drug decreases by 7%7\%7% each hour.3. The function f(x)f(x)f(x) represents the amount of drug in the patient's system after xxx hours.

STEP 2

STEP 3

Convert the percentage decrease to a decimal to find the decay rate.7%=7100=0.077\% = \frac{7}{100} = 0.077%=1007​=0.07

STEP 4

Subtract the decay rate from 1 to find the base of the exponential function.1−r=1−0.07=0.931 - r = 1 - 0.07 = 0.931−r=1−0.07=0.93

STEP 5

Write the exponential decay function using the initial amount and the base found in the previous steps.f(x)=400⋅0.93xf(x) = 400 \cdot 0.93^xf(x)=400⋅0.93x

STEP 6

Identify the correct equation from the given options that matches the exponential decay function derived in the previous step.The correct equation is:f(x)=400⋅0.93xf(x) = 400 \cdot 0.93^xf(x)=400⋅0.93x

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Question Exponential drug decay: A patient has $400 \mathrm{mg}$ of a drug decreasing by $7 \%$ per hour. Identify the correct equation: $f(x)=400 \cdot 0.93^{x}$

Assumptions
1. The initial amount of the drug in the patient's system is $400 \mathrm{mg}$ .
2. The drug decreases by $7\%$ each hour.
3. The function $f(x)$ represents the amount of drug in the patient's system after $x$ hours.

Convert the percentage decrease to a decimal to find the decay rate.
$7\% = \frac{7}{100} = 0.07$

Subtract the decay rate from 1 to find the base of the exponential function.
$1 - r = 1 - 0.07 = 0.93$

Write the exponential decay function using the initial amount and the base found in the previous steps.
$f(x) = 400 \cdot 0.93^x$

Identify the correct equation from the given options that matches the exponential decay function derived in the previous step.
The correct equation is:
$f(x) = 400 \cdot 0.93^x$