A researcher measured the flow rate, Q, of various liquids passing through a narrow capill… — ACT

A researcher measured the flow rate, $Q$ , of various liquids passing through a narrow capillary tube under constant pressure. The viscosity, $\eta$ , of each liquid and its corresponding flow rate are shown in the table.

Liquid	Viscosity ( $\eta$ , $\text{mPa}\cdot\text{s}$ )	Flow rate ( $Q$ , $\text{mL/s}$ )
Liquid 1	$2.0$	$18.0$
Liquid 2	$3.0$	$12.0$
Liquid 3	$4.0$	$9.0$
Liquid 4	$6.0$	$6.0$

Based on these results, which of the following equations best represents the relationship between the flow rate, $Q$ , and the viscosity, $\eta$ , of the liquids?

A
$Q = 36.0\eta$
B
$Q = \frac{\eta}{36.0}$
$Q = \frac{36.0}{\eta}$ Answer
D
$Q = 36.0 - \eta$

Answer

Q = \frac{36.0}{\eta}

The correct equation is the one stating that flow rate is equal to

36.0

divided by viscosity. In an inverse proportionality relationship, the product of the two variables is constant (

y \times x = k

). Multiplying the viscosity by the flow rate for each liquid in the table consistently yields

36.0

(e.g.,

2.0 \times 18.0 = 36.0

). Solving the equation

Q \times \eta = 36.0

for

Q

results in

Q = \frac{36.0}{\eta}

Step-by-Step Solution

Analyze the trend in the data table between the viscosity,

\eta

, and the flow rate,

Q

As viscosity increases from

2.0

6.0

, the flow rate decreases from

18.0

6.0

. This indicates an inverse relationship between the two variables.

Identifying whether the relationship is direct or inverse helps eliminate incorrect equation forms.

Determine if the product of the two variables is constant, which is a characteristic of inverse proportionality (

y \cdot x = k

Calculate the product

Q \times \eta

for each trial:
- For Liquid 1:

2.0 \times 18.0 = 36.0

- For Liquid 2:

3.0 \times 12.0 = 36.0

- For Liquid 3:

4.0 \times 9.0 = 36.0

- For Liquid 4:

6.0 \times 6.0 = 36.0

The product is constant at

36.0

Finding the constant of proportionality establishes the exact mathematical relationship.

Formulate the equation expressing

Q

in terms of

\eta

Since

Q \times \eta = 36.0

, solving for

Q

gives

Q = \frac{36.0}{\eta}

This matches the target variable representation requested in the prompt.

Key Concept

Inverse Proportionality

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic