Answer:
2 ( k + ( 2k +4)) = 50 cm is the required equation.
Width of rectangle = 7 cm, Length of rectangle = 18 cm
Step-by-step explanation:
The perimeter of the rectangle = 50 cm
Le the width of the rectangle = k cm
⇒The length of the rectangle = (2k + 4) cm
Now, PERIMETER OF THE RECTANGLE = 2( LENGTH WIDTH)
⇒ 2 ( k + ( 2k +4)) = 50 cm
or, 2( 3k+ 4) = 50
⇒ 6k + 8 = 50
or, 6k = 50 - 8 = 42
or, k = 42/6 = 7
⇒ k = 7 cm
Hence, the width of the rectangle = k = 7 cm
and the length of the rectangle = (2k +4) = 2(7) + 4 = 18 cm
Consider the point P where the gears meet. When the smaller gear rotates clockwise, the larger one will rotate counterclockwise.
Through one rotation of the smaller gear, P will have traveled the circumference of the smaller gear, which is
in.
At the same time, a point P' on the larger gear traverses the same distance along the larger gear's circumference. This point traces out an arc that is subtended by some angle
. The arc is as long as the smaller gear's circumference.
The measure of a circle's interior angle subtended by an arc is proportional to a complete revolution, i.e. an angular displacement of
radians:
![\dfrac{14\pi\,\mathrm{in}}{2\pi\,\mathrm{rad}}=\dfrac{8\pi\,\mathrm{in}}\theta\implies\theta=\dfrac{8\pi}7\,\mathrm{rad}\approx205.7^\circ](https://tex.z-dn.net/?f=%5Cdfrac%7B14%5Cpi%5C%2C%5Cmathrm%7Bin%7D%7D%7B2%5Cpi%5C%2C%5Cmathrm%7Brad%7D%7D%3D%5Cdfrac%7B8%5Cpi%5C%2C%5Cmathrm%7Bin%7D%7D%5Ctheta%5Cimplies%5Ctheta%3D%5Cdfrac%7B8%5Cpi%7D7%5C%2C%5Cmathrm%7Brad%7D%5Capprox205.7%5E%5Ccirc)
For part 2, we apply the same reasoning to the larger gear. In one full rotation of the larger gear, the point P' traverses the circumference
in, and so does the point P on the smaller gear.
![\dfrac{8\pi\,\mathrm{in}}{2\pi\,\mathrm{rad}}=\dfrac{14\pi\,\mathrm{in}}\theta\implies\theta=\dfrac{7\pi}2\,\mathrm{rad}=630^\circ](https://tex.z-dn.net/?f=%5Cdfrac%7B8%5Cpi%5C%2C%5Cmathrm%7Bin%7D%7D%7B2%5Cpi%5C%2C%5Cmathrm%7Brad%7D%7D%3D%5Cdfrac%7B14%5Cpi%5C%2C%5Cmathrm%7Bin%7D%7D%5Ctheta%5Cimplies%5Ctheta%3D%5Cdfrac%7B7%5Cpi%7D2%5C%2C%5Cmathrm%7Brad%7D%3D630%5E%5Ccirc)
A full rotation is 360 degrees, so the smaller gear would have rotated
times.
This problem is accompanied by a figure.
You can infere these relationships from the figure
(x-5)° = 74° => x = 74 + 5 = 79°
(x-5)° + 58° + (y-1)° = 180 ° => 74 + 58 + y-1 = 180 =>
y = 180 + 1 - 58 - 74 = 47°
Answer: x = 79°, y = 47°. This is the option d)
Your diagram implies x = 150 & r = √ ( x² + y² )