Answer:
x ∈ {-0.465, 1.014}
Step-by-step explanation:
The equation can be cast in the form f(x) = 0, and solved easily using a graphing calculator. That shows x ≈ -0.465 and x ≈ 1.014. The same calculator can iterate the roots to full calculator precision.
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The equation can be made a quadratic by the substitution ...
z = e^(2x)
Then we have ...
![2\cosh(2x)-\sinh(2x)=4\\\\2\cdot\dfrac{e^{2x}+e^{-2x}}{2}-\dfrac{e^{2x}-e^{-2x}}{2}-4=0\qquad\text{use exponential identities}\\\\2z+2z^{-1}-z+z^{-1}-8=0\qquad\text{multiply by 2, substitute z}\\\\z^2 -8z+3=0\qquad\text{multiply by z}\\\\z=\dfrac{8\pm\sqrt{(-8)^2-4(1)(3)}}{2(1)}=4\pm\sqrt{13}\\\\x=\dfrac{\ln(z)}{2}=\dfrac{\ln(4\pm\sqrt{13})}{2}\approx\{-0465133,1.014439\}](https://tex.z-dn.net/?f=2%5Ccosh%282x%29-%5Csinh%282x%29%3D4%5C%5C%5C%5C2%5Ccdot%5Cdfrac%7Be%5E%7B2x%7D%2Be%5E%7B-2x%7D%7D%7B2%7D-%5Cdfrac%7Be%5E%7B2x%7D-e%5E%7B-2x%7D%7D%7B2%7D-4%3D0%5Cqquad%5Ctext%7Buse%20exponential%20identities%7D%5C%5C%5C%5C2z%2B2z%5E%7B-1%7D-z%2Bz%5E%7B-1%7D-8%3D0%5Cqquad%5Ctext%7Bmultiply%20by%202%2C%20substitute%20z%7D%5C%5C%5C%5Cz%5E2%20-8z%2B3%3D0%5Cqquad%5Ctext%7Bmultiply%20by%20z%7D%5C%5C%5C%5Cz%3D%5Cdfrac%7B8%5Cpm%5Csqrt%7B%28-8%29%5E2-4%281%29%283%29%7D%7D%7B2%281%29%7D%3D4%5Cpm%5Csqrt%7B13%7D%5C%5C%5C%5Cx%3D%5Cdfrac%7B%5Cln%28z%29%7D%7B2%7D%3D%5Cdfrac%7B%5Cln%284%5Cpm%5Csqrt%7B13%7D%29%7D%7B2%7D%5Capprox%5C%7B-0465133%2C1.014439%5C%7D)
Answer:
I think A
Step-by-step explanation:
It is irrational.
A = π r^2 = π[3/4 cm]^2 = 9π/16 cm^2
An irrational number (Pi in this case) multiplied by a rational number (9/4) gives an irrational number.
The slope of this line is 10
Assuming that this is a right triangle, then (50 in)^2 = b^2 + (14 in)^2, according to the Pythagorean Theorem.
Then 196 in^2 + b^2 = 2500 in^2. Solving for b^2:
b^2 = (2500-196) in^2, and so b = +48 inches (answer)