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I have the answers, I just don’t know how they got to the answers.
Alright, in this case is all about using the trigonometric rules and the Pythagorean Theorem.
Just a quick review:
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent
a^2 + b^2 = c^2 (where a and b are the sides and c is the hypotenuse)
Solving problem 58:
sin(30) = 17 / c
0.5 = 17 / c
c = 17 / 0.5
c = 34
(17)^2 + b^2 = (34)^2
289 + b^2 = 1156
b^2 = 867
b = 29.4
Solving problem 59:
You can see that when a triangle has an angle of 90, 45, and 45, two of their sides are the same, so, a = b
a^2+a^2 = (12)^2
2a^2 = 144
a^2 = 72
a = 8.5
Solving problem 60:
To apply the Pythagoras Theorem, you need at least one angle of 90, to do that, and as it’s illustrated in the drawing, you can half the triangle to get the height, so the base is no longer 4, but 2.
h^2 + 4 = 16
h^2 = 12
h = 3.5
If it’s still unclear, you can check these websites:
or drop me a message.
as with any other fractions you want to add, you need the denominators (the number on the bottom) to be the same.
That’s it. There’s some different ways of writing it so it depends on what your teacher wants. I hope this helped.
12x+4=76 -> 12x=76-4 -> 12x=72 -> x=6
proof: 4(3*6+1)=76 -> 4(18+1)=76 -> 4*19=76 -> 76=76
So the trick with easy trinomials (when the x^2 has no coefficient) is to find something that multiplies into the c value (4), and adds into the b value (3), so if you’d look at the factors of 4, they’re: 1,4 and 2,2
However, in this case, it’s impossible to factor it with real solutions because this parabola has no real roots!
A quick way to see if you can ever instantly factor a trinomial whether easy or not is solve for the discriminant, which is essentially this:
so if 3^2 - 4(1)(4) = 9-16 = -7
is less than 0 (basically if it’s negative)
then the equation has no real roots because the discriminant would then have an imaginary number (when a number has a negative integer under a square root function).
This is one site which has an easy, fast explanation: http://www.mathsisfun.com/operation-order-pemdas.html
Here’s another: http://www.purplemath.com/modules/orderops.htm
I also included an explanation myself:
you don’t square root it, you cube root it. Square rooting only works for squared numbers.
a^3 = a*a*a = 1000.
You can do it with your calculator, to get that a = 10.
if we subtract 2 from each side we will be left with:
7 = -x
which is the solution to minus x. We want the solution to positive x. Here we can either add x to both sides and minus 7 from both sides OR just times both sides by minus 1. Either works to get our answer.
Don’t worry, that’s okay. All you have to do is move the x around a bit until it’s on one side.
We start out with: 9 = 2 - x
We can add an x to each side: 9 (+x) = 2 -x (+x)
Which reduces down to this: 9 + x = 2
Now we subtract 9 from both sides: 9 +x (-9) = 2 (-9)
Which reduces to: x = -7.
With a little practice, you will get much faster. It’s only a matter of knowing what to do and not getting discouraged by some negatives.
Yes! If the cost is 5 dollars/m3, but remember that 90mm = 9 cm = 0.09 m , so you only multiply:
($5 /m3)(12m2)(0.09m) = $5.4
As you can see, the units get cancelled!
there is a rule with logs that a number with the same base as it’s inside is equivalent to the insides exponent. So for example:
so in this case, if we know that the base is 10, we can convert the 1 at the end to a log_10 (10), so we’ll get this line:
Now, there’s a rule of sums with logs, where if two logs with the same bases are added, you may combine the insides and multiply them inside a single log, a visual continuation would be:
now knowing this, we can raise every single number by the exponent 10. In essence this cancels out the log function. This is the inverse of log, (10^x and log(x) are inverse)
(I know this may sound confusing but it’s similar to multiplying 1/2 by 2, to reduce the fraction, you perform the inverse function of division (multiplication).
here’s an example if you’d like:
So now if you raise everything by 10, the logs get cancelled out, and you get a simple algebraic formula:
7x + 1 = 10(x - 2)
7x + 1 = 10x - 20
-3x = -21
x = 7