The rectangle i would say is a parallelogram. Since a parallelogram has sides that are parallel which explains the name: parallelogram :) :D
Question options :
a. They should be between 64 and 76 inches tall.
b. They should be close to the height that is 95% of the mean. That is, 66.5 inches, plus or minus 2 standard deviations.
c. They should be at or below the 95th percentile, which is 74.92 inches.
d. None of the above.
Answer: a. They should be between 64 and 76 inches tall.
Step-by-step explanation:
Given the following :
Assume men's height follow a normal curve ; and :
Mean height = 70 inches
Standard deviation= 3 inches
According to the empirical rule ;
Assuming a normal distribution with x being random variables ;
About 68% of x-values lie between -1 to 1 standard deviation of the mean. With about 95% of the x values lying between - 2 and +2 standard deviation of mean. With 99.7% falling between - 3 to 3 standard deviations from the mean.
Using the empirical rule :
95% will fall between + or - 2 standard deviation of the mean.
Lower limit = - 2(3) = - 6
Upper limit = 2(3) = 6
(-6+mean) and (+6+ mean)
(-6 + 70) and (6+70)
64 and 76
The area bounded by the 2 parabolas is A(θ) = 1/2∫(r₂²- r₁²).dθ between limits θ = a,b...
<span>the limits are solution to 3cosθ = 1+cosθ the points of intersection of curves. </span>
<span>2cosθ = 1 => θ = ±π/3 </span>
<span>A(θ) = 1/2∫(r₂²- r₁²).dθ = 1/2∫(3cosθ)² - (1+cosθ)².dθ </span>
<span>= 1/2∫(3cosθ)².dθ - 1/2∫(1+cosθ)².dθ </span>
<span>= 9/8[2θ + sin(2θ)] - 1/8[6θ + 8sinθ +sin(2θ)] .. </span>
<span>.............where I have used ∫(cosθ)².dθ=1/4[2θ + sin(2θ)] </span>
<span>= 3θ/2 +sin(2θ) - sin(θ) </span>
<span>Area = A(π/3) - A(-π/3) </span>
<span>= 3π/6 + sin(2π/3) -sin(π/3) - (-3π/6) - sin(-2π/3) + sin(-π/3) </span>
<span>= π.</span>
