Prove that if p ones and q zeros are placed around a circle in an arbitrary manner, where p, q, and k are positive integers satisfying p$\geq$kq, the arrangement must contain at least k consecutive ones.

A function f has derivative

$$f ^ { \prime } ( x ) = \frac { 1 } { x ^ { 4 } + 1 }.$$

Where on the interval [1, 4] does f take on its maximum value?

State the instructions of the function in words. R(r) = 3(2r + 5)

Solve equation $s-5=3$ and check your answer.

Find the exact value of each of the remaining trigonometric functions. $\csc \theta=-4,\ \pi<\theta<\frac{3 \pi}{2}$

Simplify. Then write a word problem involving the given numbers and operation.

$$\$ 64.32 \div 16$$

Hay que buscar ayuda para la fiesta de fin de año de tu clase. ¿Qué pregunta el presidente de la clase en la reunión? 1.desear ser el anfitrión (la anfitriona) 2.conocer a un disc jockey 3.saber bailar salsa. 4.cocinar comida puertorriqueña

Answer with “converges,” “diverges” or “can’t tell.” If $\lim _{k \rightarrow \infty} a_{k}=1$, then $\sum_{k=1}^{\infty} a_{k}$ ____.

Multiply.

$$35.15\times3.7$$

List three examples each of objects that are commonly measured by mass, by volume , and by length.

Find the center and radius of a circle that has a diameter with the given endpoints.

$$(8,1) \text { and }(-4,10)$$

An airplane has an airspeed of 500 kilometers per hour (km/hr) bearing $\mathrm{N} 45^{\circ} \mathrm{E}.$ The wind velocity is 60 km/hr in the direction $\mathrm{N} 30^{\circ} \mathrm{W}.$ Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

The National Center for Education Statistics monitors many aspects of elementary and secondary education. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 31% of students reported that their mothers had graduated from college. In 2000, responses from 8368 randomly chosen students found that this figure had grown to 32%. Is this evidence of a change in education level among mothers? Write appropriate hypotheses.

Explain how a carbon-zinc dry cell produces a voltage difference between the positive and negative terminals.