Math Snap

STEP 1

What is this asking?
If the radius of a blood vessel increases, how much does the resistance of the blood vessel decrease?
Watch out!
The radius is increasing, so we expect the resistance to decrease.
Don't mix up a decrease with an increase!

STEP 2

1. Set up the equation
2. Calculate the new radius
3. Calculate the new resistance
4. Calculate the percent change in resistance

STEP 3

We're given the formula $R = \frac{kl}{r^4}$, where $k$ is a constant and $l$ is also constant.
This tells us how the resistance $R$ relates to the radius $r$.

STEP 4

The problem says the drug increases $r$ by 8%.
This means the new radius, let's call it $r'$, is the original radius $r$ plus 8% of $r$.

STEP 5

Mathematically, we can write this as:
$$r' = r + 0.08r = 1.08r$$ So the new radius $r'$ is 1.08 times the original radius.

STEP 6

Now, let's find the new resistance, which we'll call $R'$.
We can use the same formula, but with the new radius $r'$:
$$R' = \frac{kl}{(r')^4}$$

STEP 7

We know $r' = 1.08r$, so we can substitute that in:
$$R' = \frac{kl}{(1.08r)^4} = \frac{kl}{1.08^4 r^4}$$

STEP 8

We can rewrite this as:
$$R' = \frac{1}{1.08^4} \cdot \frac{kl}{r^4}$$ Notice that $\frac{kl}{r^4}$ is just the original resistance, $R$!
So,
$$R' = \frac{1}{1.08^4} R$$ $$R' \approx \frac{1}{1.3605} R \approx 0.735 R$$

STEP 9

The percent change is calculated as:
$$\frac{\textbf{New Value} - \textbf{Old Value}}{\textbf{Old Value}} \cdot 100\%$$

STEP 10

In our case, the new value is $R'$ and the old value is $R$:
$$\frac{R' - R}{R} \cdot 100\%$$

STEP 11

We know $R' \approx 0.735R$, so let's plug that in:
$$\frac{0.735R - R}{R} \cdot 100\% = \frac{-0.265R}{R} \cdot 100\% = -0.265 \cdot 100\% = -26.5\%$$

The resistance $R$ decreases by approximately 26.5%.