The propellers for a toy boat are manufactured by stamping out a rose with n petals and then bending each blade. If the manufacturer needs to program the machine to stamp out propellers composed of 5 blades with a radius of 15 mm, what two polar equations will satisfy these requirements?

Trigonometry (MindTap Course List)
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ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter8: Complex Numbers And Polarcoordinates
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**Toy Boat Propeller Manufacturing Process Using Polar Equations**

**Problem Statement:**

The propellers for a toy boat are manufactured by stamping out a rose with \( n \) petals and then bending each blade. If the manufacturer needs to program the machine to stamp out propellers composed of 5 blades with a radius of 15 mm, what two polar equations will satisfy these requirements?

---

**Solution Explanation:**

To solve this problem, we need to determine the appropriate polar equations that describe a rose curve with 5 petals and a specified radius of 15 mm.

The general form of the polar equation for a rose curve is:

\[ r = a \cdot \cos(k\theta) \]
or
\[ r = a \cdot \sin(k\theta) \]

where:
- \( r \) represents the radius,
- \( a \) represents the amplitude (maximum radius),
- \( k \) is a constant that affects the number of petals,
- \( \theta \) is the angle in radians.

Given that the propeller is composed of 5 blades:
- The number of petals \( n \) in a rose curve is determined by the value of \( k \). For \( k \) odd, the rose has \( k \) petals, and for \( k \) even, the rose has \( 2k \) petals.
- Since we need 5 blades, \( k = 5 \) (as 5 is an odd number).

Given that the radius is 15 mm, this radius is the maximum value of \( r \), and hence it corresponds to \( a = 15 \).

Thus, the two possible polar equations are:

1. \( r = 15 \cdot \cos(5\theta) \)
2. \( r = 15 \cdot \sin(5\theta) \)

These equations satisfy the requirement of having 5 petals with a maximum radius of 15 mm.
Transcribed Image Text:**Toy Boat Propeller Manufacturing Process Using Polar Equations** **Problem Statement:** The propellers for a toy boat are manufactured by stamping out a rose with \( n \) petals and then bending each blade. If the manufacturer needs to program the machine to stamp out propellers composed of 5 blades with a radius of 15 mm, what two polar equations will satisfy these requirements? --- **Solution Explanation:** To solve this problem, we need to determine the appropriate polar equations that describe a rose curve with 5 petals and a specified radius of 15 mm. The general form of the polar equation for a rose curve is: \[ r = a \cdot \cos(k\theta) \] or \[ r = a \cdot \sin(k\theta) \] where: - \( r \) represents the radius, - \( a \) represents the amplitude (maximum radius), - \( k \) is a constant that affects the number of petals, - \( \theta \) is the angle in radians. Given that the propeller is composed of 5 blades: - The number of petals \( n \) in a rose curve is determined by the value of \( k \). For \( k \) odd, the rose has \( k \) petals, and for \( k \) even, the rose has \( 2k \) petals. - Since we need 5 blades, \( k = 5 \) (as 5 is an odd number). Given that the radius is 15 mm, this radius is the maximum value of \( r \), and hence it corresponds to \( a = 15 \). Thus, the two possible polar equations are: 1. \( r = 15 \cdot \cos(5\theta) \) 2. \( r = 15 \cdot \sin(5\theta) \) These equations satisfy the requirement of having 5 petals with a maximum radius of 15 mm.
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