Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are cm in diameter and
cm high. Felicia buys cat food in cylindrical cans that are
cm in diameter and
cm high. What is the ratio of the volume of one of Alex's cans to the volume one of Felicia's cans?
Using the formula for the volume of a cylinder, we get Alex, , and Felicia,
. We can quickly notice that
cancels out on both sides, and that Alex's volume is
of Felicia's leaving
as the answer.
Shauna takes five tests, each worth a maximum of points. Her scores on the first three tests are
,
, and
. In order to average
for all five tests, what is the lowest score she could earn on one of the other two tests?
Ike and Mike go into a sandwich shop with a total of to spend. Sandwiches cost
each and soft drinks cost
each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?
The eighth grade class at Lincoln Middle School has students. Each student takes a math class or a foreign language class or both. There are
eighth graders taking a math class, and there are
eight graders taking a foreign language class. How many eight graders take only a math class and not a foreign language class?
By P-I-E, we get =
.
Solving gives us .
But we want the number of students taking only a math class,
which is A. .
Easy Easy
Point (7,2) is translated up right 2 and down 5 units.
what is the new coordinate
(9,-3)
3x3-2x2-12x+8
Answer in factored form
(x+2)(x-2)(3x-2)
Carrie has a rectangular garden that measures feet by
feet. She plants the entire garden with strawberry plants. Carrie is able to plant
strawberry plants per square foot, and she harvests an average of
strawberries per plant. How many strawberries can she expect to harvest?
D.
The area of the garden is square feet. Since Carrie plants
strawberry plants per square foot, there are a total of
strawberry plants, each of which produces
strawberries on average. Accordingly, she can expect to harvest
strawberries.
How many integers between and
have four distinct digits arranged in increasing order? (For example,
is one integer.)
C.
As in Solution 1, we find that the first two digits must be , and the third digit must be at least
. If it is
, then there are
choices for the last digit, namely
,
,
,
, or
. Similarly, if the third digit is
, there are
choices for the last digit, namely
,
,
, and
; if
, there are
choices; if
, there are
choices; and if
, there is
choice. It follows that the total number of such integers is
Ricardo has coins, some of which are pennies (
-cent coins) and the rest of which are nickels (
-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?
C.
For a positive integer the factorial notation
represents the product of the integers from
to
. (For example,
.) What value of
satisfies the following equation?
A.
We have , and
. Therefore, the equation becomes
, and so
. Cancelling the
s, it is clear that
.
What does (sin2 θ + cos2 θ)=?
1
How many factors of have more than
factors? (As an example,
has
factors, namely
and
)
7
Since , we can simply list its factors:
There are
of these; only
(i.e.
of them) don't have over
factors, so the remaining
factors have more than
factors.
cos 45 +sin 60=?
Hint: Evaluate the trig functions and combine them
Answer in simplest radical form
√2/2 + √3/2
A number is called flippy if its digits alternate between two distinct digits. For example, and
are flippy, but
and
are not. How many five-digit flippy numbers are divisible by
B. 4
Consider these two operations:
What is the value of
We can substitute ,
, and
into the functions' :
Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is
years. How many years older than Bella is Anna?
Five years ago, Bella was years old, and the kitten was
years old.
Today, Bella is years old, and the kitten is
years old. It follows that Anna is
years old.
Therefore, Anna is years older than Bella.