GEOMETRY
The area of a rectangle whose consecutive sides are in a 1:2 ratio and whose perimeter is 120.
800
if x = 4, then what is square root of (x + 1/x)?
(4 + 1/4)^(1/2) = (16+1) / 4 ]^(1/2)
= (17/4)^(1/2)
= (1/2) 17^(1/2)
= one half the square root of 17
If 1/3 of a number is 4 less than 1/2 of the number, the number is
24. Let x represent the unknown number. Since 1/3 of x is 4 less than 1/2 of x, x/3 = x/2 - 4. So, 2x = 3x - 24, giving x = 24.
IF 2, 5, 8, ... is an arithmetic progression, what is a0, d, and the 10th term?
a0 = 2 and d = 3, so an = 2 + d(n-1)
Therefore a10 = 2 + 3(9) = 2 + 27 = 29
You can also get there by calculating each row:
2
5
8
11
...
29
How many zeros in 10!
Two.
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
The only zeros come from factors of 10. There are two 10's:
10 and (5 x 2)
A string with length 2(pi) is placed along the circumference of a circle. If its central angle is pi, this is the radius of the circle.
The radius of the circle is 2.
(theta/360°)(2πr) = arc length
(π/2π)2πr = 2π
πr = 2π
r = 2
The inverse function of f(x) = 5x + 10.
Let y = f(x).
Then the inverse function of y = 5x + 10 is found by switching the x and the y: x = 5y + 10
Solving for y gives the answer:
y = (1/5)x - 2
How many intersection do these equations have:
y=x2 and y = 2x + 6
Two.
One is a parabola opening up from the origin and the other is a line that hits the y-axis above the origin.
What is the length of the other leg of a right triangle measuring 60 and 61 on its one leg and hypotenuse?
Yesterday's date was a Pythagorean Triple you should know when taking the SAT. What was it?
11
7-24-25
Which is greater: 5/7 or 10/13
10/13
Find the common denominator and compare the numerators:
common denominator is 7x13. the resulting numerators are:
5x13 = 65 and 10x7 = 70, so 10/13 is larger.
Two sides of a triangle are 3 and 5. The third side must be greater than _____ and less than _____.
2, 8
Use the triangle inequality.
In a 5 km race that starts with 1000 runners, 50 runners drop out after every 2 km.
How many runners finish the race?
5/2 = 2, remainder 1, so 2*50 = 100 runners drop out.
1000 - 100 = 900 runners finish the race.
There are 11 students in our SAT Prep class. In how many orders can all 11 students walk in ONE door one at a time?
11! = 39916800
What is the probability of flipping 10 coins in a row with all 10 landing "heads"?
(1/2)(1/2) x ... (1/2) = (1/2)^10 = 1/1024
or about 0.001
So yes: Highly unlikely!
An angle inscribed in a circle intercepts a semi circle forming a triangle. If one of the acute angles of the triangle measures 20°, what does the other acute angle measure?
Hint: Draw the figure!
70°
It is a right triangle (why?), so the two acute angles are complementary. Therefore the other acute angle must be 70°.
The measure of an inscribed angle in a circle is always one half of the intercepted arc. A semicircle measures 180°, so the inscribed angle that intercepts it must be 90°.
Think of a regular polygon with n sides. The limit as n goes to infinity is what 2-D shape?
A circle.
Archimedes used his "method of exhaustion" with infinitely thin circle sectors to convert a circle to a rectangle measuring (πr)(r) = the same area as the circle = πr2.
The line y - 2 = 3(x + 5) is moved up 8 units and to the right 6 units.
What is the new equation?
y-10 = 3(x-1)
The reference point for this point-slope version of the line is (2,-5). After the translation, the reference point is (2+8, -5+6) = (10,1). The slope does not change, so the line is now:
y-10 = 3(x-1)
What is the name of the theorem that says there are EXACTLY n solutions (some may be complex, not just real) of a polynomial of degree n?
Fundamental Theorem of Algebra
Expand (a + b)3
If you don't know how to apply Pascal's triangle, just start multiplying:
(a+b)(a+b) = ?
Then ? times (a + b) and you are there!
a3 + 3a2b + 3ab2 + b3
An 18-inch pizza is cut into the traditional 8 pieces. What is the area of one piece? You can approximate it to one decimal place.
31.8 in2
Area of pizza = (pi)r2 = (pi) 92 = 81(pi)
Divide 81(pi) by 8 to get 81(pi)/8 which is about 31.8
The sum of the interior angles of a polygon is 1260. Find the number n of sides of this polygon.
sum of interior angles = (n-2) 180°.
Why 180°: Draw all the possible triangles from a given vertex, say of a pentagon. Notice you always get two less triangles than the number of sides.
Solve: (n-2)180 = 1260.
n = 7
Two trains are on the same track 120 km from each other. One is traveling 20 km per hour and the other 40 km per hour. A bee traveling 60 km per hour is flying back and forth between the trains until they meet.
How far does the bee travel?
120 km
The trains are traveling towards each other at a combined 60 km per hour so it takes them 2 hours to meet. In 2 hours, the bee flies (60 km x 2 = 120 km.
10 years ago, I was 1/3 my Mom's age. Now, I am 1/2 her age. How old are my Mom and I now?
I am 20 and my Mom is now 40 years old.
Let m = my age, and p = my Mom's age NOW.
Then:
3(m - 10) = p - 10
2m = p
Solving for m and p gives m = 20 and p = 40.
What is the slope of the line going through these two points: (-5,-2) and (-1,4)
(-1 - -5) / (4 - -2)
A car is covering the number of a parking space. What is this number?
: |6 : 06 : 68 : 88 : ?? : 98 :
･･･････････････････････････････････････････
87
Look at this upside down!