Question

the variable f varies inversely as the square root of g. when f = 4, g = 4. jordan’s work finding the value of f when g = 100 is shown:
$f\sqrt{g} = k$
$4(4) = k$
$16 = k$
$f\sqrt{g} = 16$
$f\sqrt{100} = 16$
$10f = 16$
$f = 1.6$
what is the first error, if any, in jordan’s work?

a. he used an equation that models direct variation instead of inverse variation.

b. he incorrectly calculated the constant of variation.

c. he took the square root of the wrong variable.

d. he did not make any errors.

Explanation:

Step1: Recall inverse variation formula

If \( f \) varies inversely as \( \sqrt{g} \), the formula is \( f=\frac{k}{\sqrt{g}} \), which can be rewritten as \( f\sqrt{g}=k \)? Wait, no—wait, inverse variation: \( f \propto \frac{1}{\sqrt{g}} \), so \( f = \frac{k}{\sqrt{g}} \), multiplying both sides by \( \sqrt{g} \) gives \( f\sqrt{g}=k \). Wait, but let's check the first step. Wait, when \( f = 4 \), \( g = 4 \), so \( \sqrt{g}=\sqrt{4}=2 \). Jordan used \( 4(4)=k \), but it should be \( 4\times\sqrt{4}=k \), so \( 4\times2 = k \), so \( k = 8 \). Wait, no—wait, the first step: Jordan wrote \( f\sqrt{g}=k \), then substituted \( f = 4 \), \( g = 4 \) as \( 4(4)=k \), but \( \sqrt{g} \) when \( g = 4 \) is \( 2 \), not \( 4 \). Wait, but let's re-express the inverse variation. If \( f \) varies inversely as \( \sqrt{g} \), then \( f = \frac{k}{\sqrt{g}} \), so \( k = f\sqrt{g} \). So when \( f = 4 \), \( g = 4 \), \( \sqrt{g}=2 \), so \( k = 4\times2 = 8 \). Jordan calculated \( k = 4\times4 = 16 \), which is wrong. Wait, but let's check the options. Option A: He used direct variation instead of inverse. Wait, \( f\sqrt{g}=k \) is a constant, so that's inverse variation? Wait, no—if \( f \) varies inversely as \( \sqrt{g} \), then \( f = \frac{k}{\sqrt{g}} \), so \( f\sqrt{g}=k \), which is correct for inverse variation (product is constant). Wait, but then the error is in calculating \( k \). When \( f = 4 \), \( g = 4 \), \( \sqrt{g}=2 \), so \( k = 4\times2 = 8 \), but Jordan did \( 4\times4 = 16 \). So option B: He incorrectly calculated the constant of variation. Wait, but let's check option A again. Wait, maybe I made a mistake. Wait, inverse variation: \( y \propto \frac{1}{x} \) means \( yx = k \). So if \( f \propto \frac{1}{\sqrt{g}} \), then \( f\sqrt{g}=k \), which is correct. So the formula \( f\sqrt{g}=k \) is correct for inverse variation. Then the first error is in calculating \( k \): when \( f = 4 \), \( g = 4 \), \( \sqrt{g}=2 \), so \( k = 4\times2 = 8 \), but Jordan did \( 4\times4 = 16 \). So option B: He incorrectly calculated the constant of variation. Wait, but let's check the options again. Option A: He used direct variation instead of inverse. But \( f\sqrt{g}=k \) is inverse (product is constant). So option A is wrong. Option B: He incorrectly calculated \( k \). When \( f = 4 \), \( g = 4 \), \( \sqrt{g}=2 \), so \( k = 4\times2 = 8 \), but Jordan did \( 4\times4 = 16 \), so that's an error in calculating \( k \). So the first error is in calculating the constant of variation, which is option B? Wait, no—wait, let's re-express. Wait, the problem says "the variable \( f \) varies inversely as the square root of \( g \)". So \( f = \frac{k}{\sqrt{g}} \), so \( k = f\sqrt{g} \). So when \( f = 4 \), \( g = 4 \), \( \sqrt{g}=2 \), so \( k = 4\times2 = 8 \). Jordan's first step: \( 4(4)=k \) – he used \( g = 4 \) instead of \( \sqrt{g}=2 \), so he incorrectly calculated \( k \). So the first error is in calculating the constant of variation, which is option B? Wait, but let's check the options again. Option A: He used direct variation instead of inverse. But \( f\sqrt{g}=k \) is inverse (since \( f \) and \( \sqrt{g} \) are inversely proportional, their product is constant). So option A is incorrect. Option B: He incorrectly calculated the constant of variation. Yes, because when \( g = 4 \), \( \sqrt{g}=2 \), so \( k = 4\times2 = 8 \), but he did \( 4\times4 = 16 \). So the first error is in calculating \( k \), which is option B? Wait, no—wait, the first step Jordan did: \( f\sqrt{g}=k \), then \( 4(4)=k \). So he substituted \( g = 4 \) instead of \( \sqrt{g}=2 \), so that's an error in calculating \( k \). So the answer is B? Wait, but let's check the options again. Option A: He used direct variation instead of inverse. But \( f\sqrt{g}=k \) is inverse (product is constant). So option A is wrong. Option B: He incorrectly calculated the constant of variation. Yes, because \( k = f\sqrt{g} \), so with \( f = 4 \), \( \sqrt{g}=2 \), \( k = 8 \), but he got \( 16 \). So the first error is in calculating \( k \), so option B.

Step2: Analyze each option

- Option A: The formula \( f\sqrt{g}=k \) is correct for inverse variation (product of \( f \) and \( \sqrt{g} \) is constant), so he did not use direct variation. Eliminate A.
- Option B: When calculating \( k \), he used \( g = 4 \) instead of \( \sqrt{g}=2 \). So \( k = 4\times\sqrt{4}=4\times2 = 8 \), but he calculated \( 4\times4 = 16 \). This is an error in calculating the constant of variation.
- Option C: He took the square root of \( g = 100 \) correctly ( \( \sqrt{100}=10 \) ), so this is not the error. Eliminate C.
- Option D: He made an error, so eliminate D.

Answer:

B. He incorrectly calculated the constant of variation.