Solve the exponential equation. 3^4x-1=3^31
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The characteristics of similarity of circles is that all circles are similar
The result of the comparison between the parameters of the original and the replica of the Ferris wheel are as follows;
i) Similar Ferris wheels are similar represent similar circles,
The central angles of both Ferris wheels are equal
ii) The smaller Ferris wheel has a smaller arc length than the original Ferris wheel
iii) The equation of the replica of the Ferris wheel is x² + y² = 900
iv) Please find attached the image of the Ferris wheel
The reason for the above values is presented as follows;
Known;
The given parameters of the original Ferris wheel are;
The Ferris wheel name = Singapore Flyer
Wheel diameter = 150 m
Number of compartments = 28
Wheel circumference ≈ 471 m
Wheel area ≈ 17,662.5 m²
Central angle measure = 12.86°
Central angle measure in radians ≈ 0.224
The distance between two cars (arc length) = 16.8
Sector area of region between two cars = 630 m²
Strategy;
We show that the measure of angles in the similar circles are equal
The required parameter are;
The parameters of smaller replica of the Ferris wheel;
The diameter of the wheel, D = 60 meters
The location of the center = (0, 0)
Number of compartments = 28
Wheel circumference, C = π × D = π × 60 ≈ 188.5 m
Wheel area ≈ π × D²/4 = π × (60 m)²/4 ≈ 2827.43 m²
Central angle measure = 360/28 ≈ 12.86°
Central angle measure in radians = 2×π/28 ≈ 0.224
The distance between two cars (arc length) = C/28 = 188.5/28 ≈ 6.73 m
Sector area of region between two cars = 2827.43 m²/28 ≈ 101 m²
i) The original and replica of the Ferris wheel are similar, therefore, the conclusion that can be drawn about the central angle is that they will be equal
ii) The arc length of the smaller Ferris wheel is less than the original Ferris wheel
iii) The general equation of a circle is (x - h)² + (y - k)² = r²
Where;
(h, k) = The center of the circle = (0, 0)
r = The radius of the circle of the Ferris wheel = D/2 = 60 m/2 = 30 m
Therefore, the equation of the smaller replica of the Ferris is presented as follows;
(x - 0)² + (y - 0)² = x² + y² = 30² = 900
The equation of the replica of the Ferris wheel is x² + y² = 900
iv) The drawing of the Ferris wheel created with MS Excel is attached
Learn more about Ferris wheel here;
