The telescope at the Palomar Observatory in California has a parabolic mirror. The circle at the top of the parabolic mirror has a 200-inch diameter. An equation that approximates the parabolic cross-section of the surface of the mirror is

$$y = \frac { 1 } { 2639 } x ^ { 2 }$$

, where x and y are measured in inches. How far is the focus from the vertex of the mirror? If a point on a parabola whose vertex is at the origin is known, then the equation of the parabola can be found. For instance, if (4, 1) is a point on a parabola with vertex at the origin, then we can find the equation as follows:

$$y = \frac { 1 } { 4 p } x ^ { 2 }$$

\cdot

$$Begin with the general form of the equation of a parabola.$$

1 = $\frac { 1 } { 4 p }$ ( 4 ) ^ { 2 }\cdot

$$The known point is (4, 1). Replace x by 4 and y by 1.$$

1 = $\frac { 4 } { p }$\cdot

$$Solve for p. p=4$$

\cdot

$$p=4 in the equation$$

y = $\frac { 1 } { 4 p }$ x ^ { 2 }

$$. The equation of the parabola is$$

y = $\frac { 1 } { 16 }$ x ^ { 2 }

$$.$$

Use the unit circle to find the exact value of the trigonometric function.

$$\cos \frac { \pi } { 2 }$$

Graph each function.y = 3x2

Two friends each invested $20,000 of their own (equity) funds. Stan, being more conservative, purchased utility and manufacturing corporation stocks. Theresa, being a risk taker, leveraged the$20,000 and purchased a $100,000 condo for rental property. Considering no taxes, dividends, or revenues, analyze these two purchases by doing the following for one year after the funds were invested. (a) Determine the year-end values of their equity funds if there was a 10% increase in the value of the stocks and the condo. (b) Determine the year-end values of their equity funds if there was a 10% decrease in the value of the stocks and the condo. (c) Use your results to explain why leverage can be financially risky.

Find each quotient. $\frac { -55 } { 11 } =\text{\_\_\_\_}$

Determine whether the statements that follow are true or false, and justify your answer: There exists a real number k such that the matrix

$$\begin{bmatrix} k-2 &3\\-3 &k-2 \end{bmatrix}$$

fails to be invertible.

find approximate values of the solution of the given initial value problem at t=0.5, 1.0, 1.5, and 2.0.(a) Use the improved Euler method with h=0.025.(b) Use the improved Euler method with h=0.0125. y'=2t+e−ty,y(0)=1

A disomic product of meiosis is obtained. What is its likely origin? What other genotypes would you expect among the products of that meiosis under your hypothesis?

Calculate the integral, assuming that

$$\int _ { 0 } ^ { 5 } f ( x ) d x = 5 , \quad \int _ { 0 } ^ { 5 } g ( x ) d x = 12.$$

$$\int _ { 0 } ^ { 5 } ( f ( x ) - x ) d x$$

Solve each equation. If the equation is an identity, write identity. If it has no solution, write no solution. r/2 + 6 = 3 - 2r

Simplify.

$$b \left( b ^ { 2 } \right) \left( b ^ { 3 } \right)$$

Solve. The length of a rectangular solid aquarium is 18 in. and the width is 12 in. If the volume of the aquarium is 1836

$$in^3$$

, what is the height of the aquarium?

a. What attracted settlers to the Middle Colonies? b. What service was performed at gristmills? c. Why might enslaved Africans be able to join in rebellion more easily in the city than the country?

Simplify. $(+3) + (-14)$