2004 AMC 12B Problems: Difference between revisions
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== Problem 19 == | == Problem 19 == | ||
A truncated [[cone]] has horizontal bases with radii <math>18</math> and <math>2</math>. A [[sphere]] is tangent to the top, bottom, and lateral surface of the truncated cone. What is the [[radius]] of the sphere? | |||
<math>\mathrm{(A)}\ 6 | |||
\qquad\mathrm{(B)}\ 4\sqrt{5} | |||
\qquad\mathrm{(C)}\ 9 | |||
\qquad\mathrm{(D)}\ 10 | |||
\qquad\mathrm{(E)}\ 6\sqrt{3}</math> | |||
[[2004 AMC 12B Problems/Problem 19|Solution]] | [[2004 AMC 12B Problems/Problem 19|Solution]] | ||
Revision as of 20:13, 16 September 2008
Problem 1
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
Problem 2
In the expression 
, the values of 
, 
, 
, and 
are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?
Problem 3
If 
and 
are positive integers for which 
, what is the value of 
?
Problem 4
An integer , with 
, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of is a 7?
Problem 5
On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had Canadian dollars left. What is the sum of the digits of ?
Problem 6
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the follow- ing is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
Problem 7
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
Problem 8
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
If 
and 
with and real, what is the value of 
?
Problem 14
Problem 15
Problem 16
A function 
is defined by 
, where 
and 
is the complex conjugate of 
. How many values of satisfy both 
and 
?
Problem 17
For some real numbers and , the equation
![\[8x^3 + 4ax^2 + 2bx + a = 0\]](http://latex.artofproblemsolving.com/c/7/7/c776fa3cc39e7fb51e32ea87c738b36491d7ed49.png)
has three distinct positive roots. If the sum of the base-
logarithms of the roots is 
, what is the value of ?
Problem 18
Problem 19
A truncated cone has horizontal bases with radii 
and . A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
