The lines containing the altitudes of a triangle meet at a single point, called the orthocenter. The position of the orthocenter is determined by the measures of the angles. Construct the three altitudes for each of the following triangles. Then complete the statements in parts a-c.

b. Where does the orthocenter of a right triangle lie ?

Find values for $A, C, D, E$, and $F$ so that the graph of $A x^2+C y^2+D x+E y+F=0$ is

$a$. a pair or lines $b$. a single point

Are euglenas is single-celled organism?

Determine whether the statement is true or false. Justify your answer. The graph of the equation

$$A x^2+B x y+C y^2+D x+E y+F=0$$

can be a single point.

Consider the function

$$f ( x ) = \left( a b ^ { x } + ( 1 - a ) c ^ { x } \right) ^ { 1 / x }$$

, where a, b, and c are positive real numbers with 0<a<1. Graph f for several sets of (a, b, c). Verify that in all cases f an increasing function with a single inflection point, for all x.

A paint manufacturing company estimates that it can sell g=f(p) gallons of paint at a price of p dollars per gallon. Given that dg/

$$d \left. p \right| _ { p = 10 }$$

=-100, what can you say about the effect of increasing the price from $10 per gallon to$11 per gallon?

What does it mean when we say energy is quantized?

If $\log _{2} P=x, \log _{2} Q=y$, and $\log _{2} R=z$ write in terms of x, y, and z: $\log _{2}\left(P^{2} \sqrt{Q}\right)$

To prove Properties P1, P2, P3, and P7 of Theorem 3, let $X=\log _a M$ and $Y=\log _a N$, and give reasons for the steps listed.

Proof of P7 of Theorem 3. Let $\log _b M=R$. Then $b^R=M$, and $\log _a\left(b^R\right)=\log _a M$. Thus, $\quad R \cdot \log _a b=\log _a M$, and $R=\frac{\log _a M}{\log _a b}$. It follows that

$$\log _b M=\frac{\log _a M}{\log _a b}$$

Evaluate each expression if $a=\frac{10}{24}, b=-3 \frac{1}{8}$, and $a=\frac{10}{24}, b=-3 \frac{1}{8}$. Write the product in simplest form. bc

Find the missing properties among (P, T, v, u, h) together with x, if applicable, and give the phase of the substance.a. R-410a, T = 50 F, u = 85 Btu/lbmb. H2O, T =600 F, h=1322 Btu/lbmc. R-410a, P = 150 lbf/in.2, h = 135 Btu/lbm

Solve by completing the square.

$$x ^ { 2 } - 10 x = 24$$

Evaluate the expression. 14. 5 ³√729 - 24

Solve each inequality.

$$- 6 k > - 24$$