91Ó°ÊÓ

$$ \alpha=1728148040-140634693 \sqrt{151} $$ and $$ \beta=1728148040+140634693 \sqrt{151} $$ A direct computation (with integers) shows that \(\alpha \cdot \beta=1\). Calculate decimal representations of \(\alpha\) and \(\beta .\) Use them to compute the product \(\alpha \cdot \beta .\) Note: To twenty significant digits, \(140634693 \sqrt{151}\) equals \(1728148039.9999999997 .\) Calculators that do not record enough significant digits will give 0 for the value of \(\alpha\).

Sketch the set. \(\\{(x, y): 3 x-1<0 \leq y+2 x+5\\}\)

The lowest monthly normal temperature of Nome is \(4^{\circ} \mathrm{F}\) and occurs at the end of December \((t=12)\). The highest monthly normal temperature of Nome is \(51^{\circ} \mathrm{F}\) and occurs at the beginning of July \((t=6)\). Find a model of temperature \(T\) as a function of time \(t\) that has the form \(T(t)=\) \(b+A \sin (\omega t+\phi)\).

In Exercises \(65-68\), find a function \(f\) such that \(g \circ f=h\). \(g(x)=x^{2}+2, h(x)=x^{2}-8 x+18\)

Suppose that \(A\) and \(B\) are constants that are not both zero and that \(D\) and \(E\) are any two constants. Prove that the lines \(A x+B y=D\) and \(-B x+A y=E\) are perpendicular.

Sketch the set. \(\left\\{(x, y): x

Physical therapists recommend that ramps for people who use wheelchairs rise not more than 1 in. for each foot of forward motion. Formulate this recommendation in terms of slope. If the entrance to a certain public building is \(30 \mathrm{ft}\) from the sidewalk and the front door is \(8 \mathrm{ft}\) off the ground (up a steep flight of stairs), how can a suitable wheelchair ramp be built?

A polygon is regular if all sides have equal length. For example, an equilateral triangle is a regular 3 -gon (triangle) and a square is a regular 4 -gon (quadrilateral). A polygon is said to be inscribed in a circle if all of its vertices lie on the circle. a. Show that the perimeter \(p(n, r)\) of a regular \(n\) -gon inscribed in a circle of radius \(r\) is $$p(n, r)=2 r n \sin \left(\frac{\pi}{n}\right) $$ b. Show that the area \(A(n, r)\) of a regular \(n\) -gon inscribed in a circle of radius \(r\) is $$A(n, r)=\frac{1}{2} r^{2} n \sin \left(\frac{2 \pi}{n}\right)$$

For what values of \(y_{0}\) is the distance between the points of intersection of \(y=y_{0}\) with \(y=2 x+1\) and \(y=x+2\) equal to \(1,000,000 ?\)

Sketch the set. \(\left\\{(x, y): x