Altitudes of Obtuse Triangles

Today's question is simple geometry. It deals with triangles, altitudes, sines, and other mathematical functions and objects. The question is quite long, but its answer is not that long. I guess it's another case for the "question theorem" - the length of the question and the length of the answer are inversely proportional.

In triangle ABC, A = 65 degrees, B = 13 degrees, C = 102 degrees. A line perpendicular to AC intersects the line defined by AC at point P. The perpendicular line also passes through point B. The length of PB is 17. Find the area of triangle ABC.

Get your ruler and draw this. If you read correctly, you should draw an obtuse triangle with an altitude outside the triangle. This altitude forms two right triangles: one is outside triangle ABC and one is including triangle ABC.

Since angle BCP is the supplement of angle C, it is 180 - 102 = 78 degrees. Since the outside triangle is a right triangle, we can calculate side BC of triangle ABC:
sin 78 = 17/BC
BC = 17/sin 78 = 17.379

Well, we have one side. To find the area of ABC, we need another side, and then we can use the area formula of 1/2 * a * b * sin C. It never fails.

Now that we have BC, we have two options. Use the law of sines or use a more creative way: find the hypotenuse of triangle APB. I always vote for creativity, so let's find that side.

Angle B is 13 degrees, as given. The angle adjacent to B, CBP, is 12 degrees (the complement of 78 degrees). This makes angle ABP a total of 25 degrees. Now that we have an angle and a side in a right triangle, we can find the other sides. Let's find side AB, because it uses the cosine function, which is... the cosine function:

cos 25 = 17 / AB
AB = 17 / cos 25
AB = 18.757

Now we have everything for the area formula: we have side c (AB), side a (BC), and angle B. Let's plug them all in:
S = 1/2 * c * a * sin B
S = 36.667 sq. units.

Problem solved. Wasn't that hard, was it?

Hope you enjoyed. There are more nice things like this at Super Math Tips.
Nadav

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Inverse Quadratic Function

Today I have another question in the genre of "short question, long answer". Today's question deals with inverse functions, more accurately the inverse of a quadratic function (or more accurately, what becomes a quadratic function). This is not some simple quadratic function, but rather something more difficult to solve.

Find the inverse of y = 2x + sqrt(x)

Short, isn't it? However, the solution isn't that short.

To find the inverse, we need to substitute x and y and then solve for y. Let's do it:
x = 2y + sqrt(y)

To eliminate the square root, move the 2y to the right side and square:
x - 2y = sqrt(y)
x² - 4xy + 4y² = y

Subtract y from both sides and use the distributive property:
x² - 4xy - y + 4y² = 0
x² - y(4x + 1) + 4y² = 0

This calls for the friend of any math student: the quadratic formula. Notice that squaring may add a solution, but there is only one inverse function, so we'll need to eliminate one:
y = ((4x + 1) ± sqrt((4x + 1)² - 16x²)) / 8

Yes, not a pleasant look. Let's work with that a little so see if it gets any better:
y = ((4x + 1) ± sqrt(16x² + 8x + 1 - 16x²)) / 8
y = ((4x + 1) ± sqrt(8x + 1)) / 8

That's the best it gets, but there's still a problem: there are two possible inverse functions here, but only one is right. To find out which one is right, remember that we reached the following conclusion:
x - 2y = sqrt(y)

This means x - 2y must be non-negative. Let's plug the two possible y values to see what we get:
x - ((4x + 1) + sqrt(8x + 1)) / 4 = 4x / 4 - ((4x + 1) + sqrt(8x + 1)) / 4
= (4x - ((4x + 1) + sqrt(8x + 1))) / 4 = (-1 - sqrt(8x + 1)) / 4
In one word: negative. Not good.

Let's plug the other one now to see if it works:
x - ((4x + 1) - sqrt(8x + 1)) / 4 = 4x / 4 - ((4x + 1) - sqrt(8x + 1)) / 4
= (4x - ((4x + 1) - sqrt(8x + 1))) / 4 = (-1 + sqrt(8x + 1)) / 4

As you can clearly see, sqrt(8x + 1) > 0 if x > 0, just like the domain of the original function. That means the inverse of the original function is:
y = ((4x + 1) - sqrt(8x + 1)) / 8

Here is a little confirmation test:
Plug 9 for x in the original function. You get y = 2*9 + sqrt(9) = 21. Now plug 21 in the inverse function and see if it gives back 9:
y = ((4 * 21 + 1) - sqrt(8 * 21 + 1)) / 8
y = (85 - sqrt(168 + 1)) / 8
y = (85 - sqrt(169)) / 8
y = (85 - 13) / 8
y = 72 / 8 = 9

Yes, it is the inverse.

You can find many more interesting math tricks like that on super math tips.

Hope you liked it.
Nadav

nadavs

Wire Optimization

Finally, after waiting for a good question, it finally came: an optimization question. This one is from Yahoo Answers, and believe me, it's very good.

A 6 meter long wire is cut into 12 pieces. From these pieces, eight have the same length and the other four also have an equal length. These pieces form a frame of the box. How long should each piece for the box to have a maximum volume?

First, we should define ourselves some variables, otherwise we'd be lost.

Let x be the length of one of the 8 pieces. That means the total length of the 8 equal pieces is 8x. That means the other four pieces have a total length of 6 - 8x, and 3/2 - 2x meters each.

Since these wires form a box, the sides of the box have lengths of x, x, and 3/2 - 2x. That makes the volume of the box x * x * (3/2 - 2x), or 3x²/2 - 2x³.

To find the maximum value, we need to differentiate the volume function and set it to zero. Then we need to find the values that we get from solving the equation, plug them in the second derivative, see which one is a maximum and say "Problem Solved".

So:
V = 3x²/2 - 2x³
V' = 3x - 6x²
0 = 3x - 6x²
3x(1 - 2x) = 0
x = 0, 1/2

Now, let's find the second derivative and see when it's negative, so we get a maximum:
V'' = 3 - 12x
V''(0) = 3 - 0 = 3 > 0 - minimum
V''(1/2) = 3 - 12 * 1/2 = 3 - 6 = -3 < x =" 1/2" 1 =" 1/2">3 = 1/8 m³

Problem solved.
Nadav

nadavs

Chairs, Tables, and Ratios

Today I have a very easy question, but once again, two solutions. They are different in their approach, and anyone can choose what works best for him.

The ratio of the price of a table to the price of a chair is 5:3. A table costs $500 more than a chair. How much does each cost?

First, here is one answer (not mine):

Let x be the price of chairs.
That means x + 500 is the price of tables. From the ratio, we can solve the following equation and get each price:
5/3 = (x + 500) / x
5x = 3x + 1500
2x = 1500
x = 750

That makes tables cost $1250. And indeed, when you figure out the ratio, 1250:750 = 25:15 = 5:3.

However, there is another way to answer this question (my way):
Since the ratio is 5:3, you can also call it 5x:3x, where 5x is the price of a table and 3x is the price of a chair. Now we can solve a simpler equation:
5x = 3x + 500
2x = 500
x = 250

Notice that x is not a price of a chair or a table, it's just the unit that gives the price of each. That means the price of a table is $250 * 5 = $1250 and chairs cost $250 * 3 = $750.

This is another way of solving simple math questions using creative thinking.

Hope you liked it,
Nadav

nadavs

Two Way Limits

It does not happen often, but when it does, it's simply beautiful: a question with two completely different yet right answers. I saw such question on Yahoo Answers today. Although I didn't answer it, the two different answers were so interesting and different in their approach, I had to bring it here.

The question is quite simple for people who know limits, but the two different answers are just amazing.

Find the limit:
lim_{x -> 4} (3 - (5 + x)^1/2) / (1 - (5 - x)^1/2)

As you can clearly see, plugging x = 4 gives a zero in the denominator, which is why the question asks for a limit. You can plug in numbers and get a rough estimation, but we want a definite answer.

The first method is using L'Hopital's rule. This rule says that in order to find lim_{x -> a} f(x)/g(x), you can also find lim_{x -> a} f'(x)/g'(x). All we need now is to differentiate the numerator and denominator and see what we get:
lim_{x -> 4} (-1/2 * (5 + x)^-1/2) / (1/2 * (5 - x)^-1/2)

The halves cancel, and by using the law that says aⁿ / bⁿ = (a/b)ⁿ, we can show that:
lim_{x -> 4} -((5 + x) / (5 - x))^-1/2

Now plug in 4 and you will get -((5 + 4)/(5-4))^-1/2
Which is -9^-1/2 = -1/3

This is one very good solution, and most people who know calculus would choose that. However, there is another solution to this problem. To do that, multiply the numerator and denominator by the conjugate of the denominator:
lim_{x -> 4} (3 - (5 + x)^1/2) / (1 - (5 - x)^1/2) * (1 + (5 - x)^1/2) / (1 + (5 - x)^1/2)

Using the law that says (a + b)(a - b) = a² - b² and the distributive property, we can conclude that:
lim_{x -> 4} (3 - (5 + x)^1/2) * (1 + (5 - x)^1/2) / (1 - (5 - x))
lim_{x -> 4} (3 - (5 + x)^1/2) * (1 + (5 - x)^1/2) / (x - 4)

Now multiply both parts of the fraction by the conjugate of the original numerator:
lim_{x -> 4} (3 - (5 + x)^1/2) * (1 + (5 - x)^1/2) / (x - 4) * (3 + (5 + x)^1/2) / (3 + (5 + x)^1/2)
lim_{x -> 4} (9 - (5 + x)) * (1 + (5 - x)^1/2) / ((x - 4) * (3 + (5 + x)^1/2))
lim_{x -> 4} (4 - x) * (1 + (5 - x)^1/2) / ((x - 4) * (3 + (5 + x)^1/2) )

That is really nice. (4 - x) / (x - 4) = -1, so we can cancel out two terms and turn them into a nice little minus sign:
lim_{x -> 4} -(1 + (5 - x)^1/2) / (3 + (5 + x)^1/2)

Now we can safely plug 4 for x and we get:
-(1 + (5 - 4)^1/2) / (3 + (5 + 4)^1/2)
-(1 + 1^1/2) / (3 + 9^1/2)
-(1 + 1) / (3 + 3)
-1/3

Again, we get -1/3 as an answer.

Math can be very easy if you think outside the box. Try doing it as often as you can.
Nadav

nadavs

Multiplication and Remainders

Today's question is a very nice question I found on Yahoo Answers. There was a trigonometry question I was planning for today, but this question is more unique (and besides that, this blog has already many trigonometry questions).

When an integer N is divided by D, it gives a remainder of 7. When N is multiplied by 5 and divided by D, it gives a remainder of 10. What is D?

At first, you may think "I need N to do that". Well, if you had N, the solution would be obvious. That's why we need some creative thinking to find what this number D is.

First, we know that D must be greater than 10. If a/b gives a remainder of m, then b must be greater than m (if you divide something by 7, you can't get a remainder of 8, since that means the result of division must be greater by 1).

Now, since N/D gives a remainder of 7:
N = xD + 7

Also, we know that 5N/D gives a remainder of 10:
5N = yD + 10

Where x and y are integers.

Now multiply the first equation by 5:
5N = 5xD + 35
And equate it to the second one:
5xD + 35 = yD + 10
yD - 5xD = 25
D(y - 5x) = 25
D = 25 / (y - 5x)

Since D must be an integer, 25 must be divided by one of its factors: 1, 5, or 25. However, dividing it by 25 or 5 will give 1 or 5, which is less than 10. For this reason, D must be 25.

You can now take any number for x and create N. For example, let's see what happens when x = 3:
N = 3*25 + 7
N = 75 + 7 = 82
N/25 = 3 with remainder 7 (not surprising, as we went from the definition backwards).

Now multiply this N by 5:
5N = 410
Divide by 25:
410 / 25 = 16 with remainder 10

Nice, isn't it?
Have a great week,
Nadav

nadavs

Percentages and Copy Machines

Today's question is relatively easy, but very unique. It does not require complex mathematical calculations, but rather a simple mathematical thinking.

A copy machine has 7 buttons to increase or decrease the area of the photocopied area. They are 50%, 75%, 80%, 100%, 120%, 125%, and 150%. Which buttons can be eliminated from the machine in a way that saves the machines original area choices. Notice: when two buttons are pushed consecutively, the effect of the second button is on the area that remains after the first button (for example, two pushes on 50% will photocopy 25%).

It seems easy, and it really is. All you have to do is turn these percents into fractions and see how you can multiply them to get the others.

50% = 1/2
75% = 3/4
80% = 4/5
100% = 1
120% = 6/5
125% = 5/4
150% = 3/2

50% cannot be created by any combination, so it must remain in the machine.
75% = 3/4 = 1/2 * 3/2
80% cannot be created by any combination, so it also stays in the machine.
100% = 1 = 5/4 * 4/5
120% = 4/5 * 3/2
125% is another number that cannot be created from others.
150% is a component in 75% and 120%, so it can't go away.

However, this question has another answer, where 150% = 6/5 * 5/4, and then 120% stays in the machine and 150% goes out.

Easy, isn't it?
Nadav

nadavs