precalc warm ups

Honors Precalc Warm-Ups

Use a calculator to approximate the following:

What does this expression represent?

If r is a real number, either positive,

negative, or zero, which of the

following is always greater than r?

Why?

Six students each try to guess the

number of pennies in a jar. The six

guesses are 52, 59, 62, 65, 49, and

42. One guess is 12 away and the

other guesses are 1, 4, 6, 9, and 11 other guesses are 1, 4, 6, 9, and 11

away. How many pennies are in

the jar?

Graph the function (without a

calculator):

f(x) = | x + 1 | + | x – 1|f(x) = | x + 1 | + | x – 1|

Each of the circles has a 10 inch radius.

ABCD is a square inscribed in a semi-

circle; MNPQ is a square inscribed in a

circle. What is the ratio of the area of

ABCD to MNPQ?

If a circle and a square have the same

perimeter, which of the following

statements are true?

A. Their areas are equal.

B. The area of the circle is greater than the

area of the square.area of the square.

C. The area of the square is greater than

the area of the circle.

D. The area of the circle is π times the area

of the square.

Draw an example of an octagon

with:

A. One line of symmetry

B. Two lines of symmetry

C. Four lines of symmetryC. Four lines of symmetry

D. Eight lines of symmetry

A vertical line divides the triangle

with vertices (0,0), (1,1) and (9,1)

into two regions of equal area.

What is the equation of the vertical

line?

The radius of a cylinder is 8 inches.

The height of the cylinder is 3

inches. What number of inches

can be added to either the radius can be added to either the radius

or the height to give the same

positive increase in volume?

A circle with area 9π is inscribed in

an equilateral triangle. What is the

area of the triangle?

Let ΔABC be a right triangle with

AC as hypotenuse. Let CAB be a

circle of diameter AB. Let CBC be a

circle of diameter BC. Let CAC be a circle of diameter BC. Let CAC be a

circle of diameter AC.

Show that

area CAB + area CBC = area CAC

What is the 500th fraction in the

sequence?sequence?

What is the nth fraction in the

sequence?

The point (3, 5) is the top vertex of

an isosceles triangle with base

parallel to the x-axis. The triangle

has an area of 12. Find the has an area of 12. Find the

coordinates of the other two

vertices.

If the circumference of a pipe is

increased from 20 inches to 25

inches, what is the increase in the inches, what is the increase in the

radius?

An 8 x 4 rectangle is centered at

(-3, 7). What are the coordinates

of the four corner points of the

rectangle is the longer side is

parallel to the y-axis?

In this figure, MNOP is a square

and DAN is an equilateral triangle.

If the area of MNOP is one square

inch, what is the area of DAN in

square inches?square inches?

Given a regular octagon with sides

of length 8, find the distance

between opposite sides.

Given f(x) = ax2 + bx + 5. Find a and

b such that f(x + 1) – f(x) = 8x + 3b such that f(x + 1) – f(x) = 8x + 3

A wheel has an outside diameter of

6 feet. How many revolutions of

the wheel are required to cause a the wheel are required to cause a

point on the rim to go one mile?

Two cars start together and move

in the same direction around a

circular race track of length 999

meters. The cars meet for the first

time 37 minutes after they start. time 37 minutes after they start.

Determine the speed of each car if

the speed of the first car is 4 times

the speed of the second.

A rectangular box measures 4 feet

by 4 feet by 3 feet. Will a 6-foot

pole fit inside?pole fit inside?

Justify your answer!

Solve the problems below. Each

letter represents a unique digit.

Letter values differ in the two

problems.

The graph of a parabola that is

symmetric about a vertical line

contains the points (1, 4), (3, 18),

and (0, 3). Find the equation of and (0, 3). Find the equation of

the parabola.

Describe and draw the solid that

results from revolving a right results from revolving a right

triangle about one of its legs.

The odometer of the family car

showed 15,951 miles. The driver

noticed that this number was

palindromic, that is, it read the same

forward or backward. Two hours later, forward or backward. Two hours later,

the odometer showed a new

palindromic number. How fast was the

car traveling in those two hours?

The function f(x) = x3 - 6x2 + 9x

has a relative extremum at x = 3. Is has a relative extremum at x = 3. Is

it a maximum or a minimum?

ABCD and EFGH are squares, each

with a side of length 12. E is the

center of ABCD. If the distance BJ

is 3, find the area of EJCK.

A fox notices a rabbit that is 30

meters away. The fox’s leap is 2

meters whereas the rabbit’s leap is

1 meter. If the fox can take 3 leaps 1 meter. If the fox can take 3 leaps

in the time that the rabbit can take

2 leaps, what distance will the fox

have to cover to catch the rabbit?

Find the points of intersection of

the two graphs given by the

equations 4x2 + y2 = 100 and equations 4x + y = 100 and

9x2 – y2 = 108

Which is larger?

or

n and k are integers. What is the

smallest value for n such that

n3 – n is not divisible by k for all 1 ≤ n3 – n is not divisible by k for all 1 ≤

k ≤ n?

Find the volume of a sphere

circumscribed about a 4 cm by 4

cm by 4 cm cube.cm by 4 cm cube.

Color the vertices of the given figure

with as many colors as needed such

that no two adjacent vertices are

colored the same color. What is the

minimum number of distinct colors minimum number of distinct colors

needed?

If (2137)753 is multiplied out, the

ones digit in the final product will ones digit in the final product will

be ________.

If the length of the diagonal of a

square is y + w, what is the area of square is y + w, what is the area of

the square?

Three circles with centers (-2, 3),

(0, 4), and (4, 6) are all tangent to

each other at the point (2, 5). Find each other at the point (2, 5). Find

an equation for each circle.

An equivalent expression for

is:is:

The medians of a triangle meet at

the centroid. Find the centroid of

triangle ABC if A is (1, -2), B is triangle ABC if A is (1, -2), B is

(-2, 4), and C is (1, 6)

A tennis ball is served from a

height of 7 feet to just clear a net 3

feet high. If it is served from a line

39 feet behind the net and travels

in a straight path, how far from the in a straight path, how far from the

net does the ball first hit the

ground?

For what values of x will the graph

of y = x2 – x – 2 be below the graph

of y = x + 1?of y = x + 1?

The dot product of two vectors is

defined as (a1, a2, a3 )(b1, b2, b3 ) =

a1 b1 + a2 b2 + a3 b3. a1 b1 + a2 b2 + a3 b3.

Find the dot product of (7, 6, 3)

and (2, -1, 6).

Suppose a cable fits tightly around

the equator of the earth. If an

additional piece is to be spliced

into the cable so that it can be

raised three meters above the raised three meters above the

earth at all points, how much

additional cable will be needed?

The altitudes of a triangle are

concurrent at the orthocenter.

Find the orthocenter of triangle

ABC if A is (-2, 2), B is (4, 3), and C ABC if A is (-2, 2), B is (4, 3), and C

is (3, -2)

If f(x) is the inverse of g(x) under

function composition, what is the

relation of the graph of f(x) to the relation of the graph of f(x) to the

graph of g(x)?

If [a] denotes the greatest integer

less than or equal to a, what is the

value of [a] + [-a] when a is a real value of [a] + [-a] when a is a real

number?

Solve for x.

sin 2x = 2 cos xsin 2x = 2 cos x

When an ice cube melts until its

sides are one third their original

length, what fractional part of the length, what fractional part of the

ice cube has melted away?

Let A(-3, 1), B(2, 3), and C(-2, 5) be

three points. If line CX is

perpendicular to line AB at the perpendicular to line AB at the

point X, find the coordinates of X.

How far apart are the two circles

x2 + y2 + 4x – 12y + 31 = 0 and

x2 + y2 - 10x – 2y + 10 = 0?

What are the x- and y- intercepts

of the hyperbola given by the

equation equation

The current in an electrical circuit

follows the equation l = 5(sin 8.72t)

where t is the time in seconds and l

is the current in amperes. What is is the current in amperes. What is

the maximum value of the current

in this circuit?

How does cos x vary as x goes from

π to ?π to ?

If a > 1 and 0 < x < y, which is

larger, log x or log y?larger, loga x or loga y?

Two high school classes took the

same math test. One class of 20

students made an average grade of

80%. The other class had an

average grade of 70%. If the average grade of 70%. If the

combined average was 74%, how

many students were in the second

class?

If f(x) is any periodic function with

period p, then

A. m · f(x) has period __________A. m · f(x) has period __________

B. f(x + m) has period __________

C. f(mx) has period ___________

A regular hexagon is inscribed in a

circle of radius 10 inches. What is circle of radius 10 inches. What is

the area of the hexagon?

The location of a cannon is given

coordinates (0, 100). The cannon

fires a projectile which follows the

path . path .

for x ≥ 0. What are the coordinates

of the point where the projectile

hits the positive x-axis?

A parachutist will land somewhere

in the square. What is the

probability that she will land in the

shaded area?

A box contains 5 red marbles and 3

green marbles. Three marbles are

selected one after the other

without replacing them as they are without replacing them as they are

selected. What is the probability

that the third marble selected is

green?

A kite escaped the hands of a child.

The little boy watched as the kite rose

200 m, traveled 1000 m due

northwest, dropped 100 m, and

continued to travel 500 m northeast. continued to travel 500 m northeast.

Then it turned and traveled 1000 m to

the southeast and finally dropped 100

m. How far was the kite from its

starting place?

Simplify the continued fraction:

Solve for x.

log (x – 5) + log (x + 1) = 2log4 (x – 5) + log4 (x + 1) = 2

A circular race track has a 50-meter

radius for one running lane and a

52-meter radius for the second

running lane. Two runners are running lane. Two runners are

beginning a race. How much of a

headstart should the outside

runner be given?

How many pairs (a, b) of non-zero

real numbers satisfy the equation real numbers satisfy the equation

?

Solve for x.

On a square piece of paper, draw

the largest possible circle and cut it

out. Inside the circle, draw the

largest possible square and cut it largest possible square and cut it

out. The final square is what

fraction of the original piece of

paper?

Four girls bought a bike for $60. The

first girl paid one half of the sum of the

amounts paid by the other girls; the

second girl paid one third of the sum

of the amounts paid by the other girls;

the third girl paid one fourth of the

sum of the amounts paid by the other

girls. How much money did the fourth

girl pay?

What is the area of the triangle

lying between the coordinate axes

and the line 3x + 8y – 24 = 0?and the line 3x + 8y – 24 = 0?

A ladder is 16 feet long. The top of

the ladder rests against a window

of a two-story house that is built

on level ground. The ladder makes

an angle between 700 and 850 an angle between 700 and 850

with the ground. What are the

minimum and maximum heights

from the ground to the window?