An engineering group uses a mathematical model to estimate the theoretical power output, P… — ACT

An engineering group uses a mathematical model to estimate the theoretical power output, $P$ (in watts, $\text{W}$ ), of a wind turbine. The model is given by the following equation:

P = \frac{1}{2} \pi \rho r^2 v^3

where $\rho$ represents the air density (in $\text{kg/m}^3$ ), $r$ represents the turbine blade length (in meters, $\text{m}$ ), and $v$ represents the wind speed (in $\text{m/s}$ ). Match each proposed modification of the turbine's operating parameters on the left to its corresponding effect on the theoretical power output ( $P$ ) on the right.

Doubling the blade length ( $r$ ) while keeping wind speed ( $v$ ) and air density ( $\rho$ ) constantThe power output increases by a factor of 4
Doubling the wind speed ( $v$ ) while keeping blade length ( $r$ ) and air density ( $\rho$ ) constantThe power output increases by a factor of 8
Halving the wind speed ( $v$ ) and doubling the air density ( $\rho$ ) while keeping blade length ( $r$ ) constantThe power output decreases to $\frac{1}{4}$ of its original value
Tripling the blade length ( $r$ ) and halving the wind speed ( $v$ ) while keeping air density ( $\rho$ ) constantThe power output is multiplied by $\frac{9}{8}$

Answer

Doubling blade length increases power by a factor of 4; doubling wind speed increases power by a factor of 8; halving wind speed and doubling air density decreases power to 1/4 of its original value; tripling blade length and halving wind speed multiplies power by 9/8.

Each modification is correctly matched by substituting the factor changes into the scaling formula derived from the model equation:

P_2/P_1 = (\rho_2/\rho_1)(r_2/r_1)^2(v_2/v_1)^3

Step-by-Step Solution

Analyze the proportionalities in the wind turbine power model

P = \frac{1}{2} \pi \rho r^2 v^3

Determine that power

P

scales with

\rho^1

r^2

, and

v^3

This establishes the scaling factors for each individual variable in the model.

Calculate the scaling factor for doubling

r

(2)^2 = 4

Matching the first modification to an increase in power output by a factor of 4.

Blade length is squared in the model, so doubling it results in a four-fold increase.

Calculate the scaling factor for doubling

v

(2)^3 = 8

Matching the second modification to an increase in power output by a factor of 8.

Wind speed is cubed in the model, so doubling it results in an eight-fold increase.

Calculate the combined scaling factor for halving

v

and doubling

\rho

2 \times (0.5)^3 = 2 \times 0.125 = 0.25

Matching the third modification to a decrease in power output to 1/4 of its original value.

Air density is linear and wind speed is cubed, leading to a net factor of 1/4.

Calculate the combined scaling factor for tripling

r

and halving

v

(3)^2 \times (0.5)^3 = 9 \times 0.125 = 9/8

Matching the fourth modification to a power output multiplied by 9/8.

Blade length squared times wind speed cubed yields a factor of 9/8.

Key Concept

Analyzing scaling relationships and proportionalities in mathematical equations representing scientific models.

Estimated Time:2m 0s

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