# Geometry – circles

Q: The area of a 45 degree sector of circle M is 100 square units. What is the area of circle M?

Explanation:  The 45° sector is exactly 1/8 of the circle. (45/360 = 1/8)

100 = 1/8 of the area of the whole circle.

The area of the whole circle is 800.

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# Algebra 1, 2 – systems of equations

Two cubes and one pyramid weigh 9 lb. Seven cubes and two pyramids weigh 24 lb. All cubes weigh the same, and all pyramids weigh the same. What is the weight of one pyramid?

Explanation:  Let x be the weight of a cube.  Let y be the weight of a pyramid.

The first sentence becomes: 2x + y = 9.

The second sentence is: 7x + 2y = 24.

Solve this as a system of equations by any method you choose… substitution, elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program ‘CRAMER’.

x = 2, y = 5

‘y’ is the weight of one pyramid, so the answer is 5.

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# Geometry – prism

How many vertices does a rectangular prism have?

If you have trouble drawing it or visualizing it, consider this instead:  A prism has two identical bases.  All of the vertices are along the edges of the two bases.  Since a rectangular prism has rectangles for bases, each base will have 4 vertices.  Two rectangular bases will have 8 vertices total.

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# SAT – counting and ordering

Q: You are 18th in line and your friend is 34th in line. How many people are in line between you and your friend?

No, not 16! Do 34-18=16, but then subtract one. Why? The 16 includes either the person you started with (you) or the person you ended with (your friend), and you don’t want to include either person in the count.

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# Algebra 2 – solving quadratic equations

Q: The length of a rectangle is 4 cm longer than the width. If the rectangle’s area is 45 cm², what is the width?

Explanation:  Call the width x.  Then, the length is x+4.

Area of a rectangle = (length)(width).

45 = x(x+4)

45 = x² + 4x

0 = x² + 4x – 45

Now, factor:

0 = (x + 9)(x – 5)

x = -9 or x = 5

Since a length can’t be negative, discard x = -9.  The only solution is x = 5.  Since ‘x’ is what we called the width, it is the answer to the question.  Just put the appropriate unit on it: 5 cm.

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# SAT – Probability

Using a standard pair of dice, what is the probability of rolling two numbers with a sum of 8?

5/36

Explanation:  6 x 6 = 36 ways you can roll two dice.  Now, find the ways you can get a sum of 8:

2+6, 6+2, 3+5, 5+3, 4+4

That’s 5 ways.  So, the probability is 5/36.

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# SAT – Venn Diagrams

Q: Of the 20 students in the classroom, 14 are taking Calculus, 12 are taking Physics, and 4 are taking neither Calculus nor Physics. How many students are taking both Calculus and Physics?

Use a Venn Diagram.  Assume for a moment that there is no overlap between Calculus and Physics.  Then, there would be 14 + 12 = 26 students taking either Calculus or Physics.  Add the 4 students taking neither, and that would be a total of 30 students.  The question states that there are only 20 students in the class.  So there must be some overlap after all.  How much?  30 – 20 = 10 students taking both Calculus and Physics.  Put 10 in the overlapping region.  To fill out the rest of the diagram: 14 – 10 = 4 students in Calculus but not Physics.  12 – 10 = 2 students in Physics but not Calculus.
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# Geometry – Triangles

Q: In triangle ABC below, what is sin C ?

Explanation:  Use the Pythagorean Theorem or the 3-4-5 Pythagorean triple to find out that the hypotenuse is 10.
Then, sin C = (opposite)/(hypotenuse) = 8/10 = 4/5.