honors 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

away. How many pennies are in the jar?

Graph the function (without a calculator):

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.

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 symmetryB. Two lines of symmetryC. Four lines of symmetryD. 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

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 AC. Show that

area CAB + area CBC = area CAC

What is the 500th fraction in the 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

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 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?

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 + 3

A wheel has an outside diameter of 6 feet. How many revolutions of 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.

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?

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

the parabola.

Describe and draw the solid that 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, 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 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 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

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 ≤ k ≤ n?

Find the volume of a sphere circumscribed about a 4 cm by 4

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 needed?

If (2137)753 is multiplied out, the 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

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 an equation for each circle.

An equivalent expression for

is:

The medians of a triangle meet at the centroid. Find the centroid of

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 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?

The dot product of two vectors is defined as (a1, a2, a3 )(b1, b2, 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 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 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

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

number?

Solve for x.sin 2x = 2 cos x

When an ice cube melts until its sides are one third their original

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 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

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 the maximum value of the current

in this circuit?

How does cos x vary as x goes from π to ?

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

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 __________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

the area of the hexagon?

The location of a cannon is given coordinates (0, 100). The cannon

fires a projectile which follows the 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

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. 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.log4 (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

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

?

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

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?

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 with the ground. What are the

minimum and maximum heights from the ground to the window?