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Chapter 0: Problem 11

Experiments conducted by A.J. Clark suggest that the response \(R(x)\) of a frog's heart muscle to the injection of \(x\) units of acetylcholine (as a percent of the maximum possible effect of the drug) can be approximated by the rational function $$R(x)=\frac{100 x}{b+x} \quad x \geq 0$$ where \(b\) is a positive constant that depends on the particular frog. a. If a concentration of 40 units of acetylcholine produces a response of \(50 \%\) for a certain frog, find the response function for this frog. b. Using the model found in part (a), find the response of the frog's heart muscle when 60 units of acetylcholine are administered.

Short Answer

Expert verified
The response function for the specific frog is \(R(x) = \frac{100x}{40 + x}\). For 60 units of acetylcholine, the response of the frog's heart muscle is \(60 \%\) of the maximum possible effect.

Step by step solution

01

Find the value of constant \(b\) for the specific frog

We know that the response function is given by: $$ R(x) = \frac{100x}{b + x} $$ We are given that when \(x = 40\) units of acetylcholine, the response is \(50 \%\), which means \(R(40) = 50\). We can plug in these values into the response function: $$ 50 = \frac{100 \cdot 40}{b + 40} $$ Now we need to solve this equation for \(b\).
02

Find the response function for this frog

Multiplying both sides of the equation by \(b + 40\) and then dividing both sides by 50 to get: $$ b + 40 = \frac{100 \cdot 40}{50} $$ $$ b + 40 = \frac{4000}{50} $$ Simplifying the right-hand side gives: $$ b + 40 = 80 $$ Now, we can subtract 40 from both sides to get the value of \(b\): $$ b = 80 - 40 $$ $$ b = 40 $$ Now that we have the value of \(b\) for this specific frog, we can find the response function by plugging the value of \(b\) into the original function: $$ R(x) = \frac{100x}{40 + x} $$
03

Calculate the response for 60 units of acetylcholine

Now, we simply plug in the given amount of acetylcholine (60 units) into the response function to find the response of the frog's heart muscle: $$ R(60) = \frac{100 \cdot 60}{40 + 60} $$ $$ R(60) = \frac{6000}{100} $$ Dividing numerator by the denominator, we get: $$ R(60) = 60 $$ So, when 60 units of acetylcholine are administered, the response of the frog's heart muscle is \(60 \%\) of the maximum possible effect.

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