Let h be the height of the wall in feet and x the length of the ladder, also in feet.
First we have: x = h + 2.
When the top of the ladder is placed against the top of the wall a right triangle forms, with x the hypotenuse and h the vertical leg.
Now the distance from the base of...
If you know how to factor it, you're more than half way there.
y2 + 3y + 2 can be factored as (y+2)(y+1) so the equation becomes:
(y+2)(y+1) = 0
Now you have a product of two real numbers equal to zero. This happens if and only if one of the factors is zero, that...
For the first step we use (ab)n = anbn.
For the second step we use (an)m = anm.
(4y3)2 = 42(y3)2 = 16y6.
3x - 2y = -1
82x + 5y = -1
To eliminate the y variable, we can multiply the first equation by (5/2), obtaining the system:
(15/2) x - 5y = -5/2
82x + 5y = -1
Adding the two equations:
[(15/2)+82] x = -5/2 - 1 = -7/2
Multiplying this equation by 2:
To prove an identity like this, it's convenient to work both sides and arrive at the same answer.
First the left hand side, using the definition of sinh:
sinh2x = ((ex - e-x)/2)2 = (ex - e-x)2/4 = ((ex)2 - 2exe-x + (e-x)2)/4
= (e2x + e-2x - 2)/4
Now for the...
There's no real number x that satisfies that equation. The square of a real number is always a non-negative number, so (x+4)2 must be positive or zero, it cannot be a negative number like -13.
Hope this helps!
x2 + y2 + 2x - 8y + 1 = 0
We have to complete the squares. Rewrite the equation:
x2 + 2x + y2 - 8y + 1 = 0
for the x part we have to add 1 to both sides of the equation and for the y part we have to...
When the pole cracked, a right triangle formed as follows: one side is the part of the pole that stands straight (call it a), the hypotenuse (call it h) is the part that fell, the upper part, and the other side is the ground from the pole to where the broken part hit (call it b = 10 meters).
To condense is the opposite of to expand. Here you use properties of logarithms.
ln(4xy2) - 2ln(x2y) = ln(4xy2) - ln( (x2y)2 ) = ln(4xy2) - ln(x4y2 ) =
= ln[(4xy2)/(x4y2)] = ln(4/x3)
That's it! Make sure you know which property was used at each step...
Define T: quarts of the 32% solution and S: quarts of the 78% solution, to be used.
Since the total is 24 quarts, we have T + S = 24.
The content of alcohol in the mix will be: 0.32 T + 0.78 S, and it should yield 50% of the 24 quarts, i.e. 12 quarts, so
0.32 T + 0.78 S =...
2(x+1)(x-3)2 -3(x+1)2 (x-3)
First you have to factor out common terms with the smallest exponents, in this case (x+1) appears with exponents 1 and 2, and (x-3) appears also with exponents 2 and 1, so we factor each of them with exponent 1
(x+1)(x-3) [ 2(x-3) - 3(x+1)...
For this question it's useful to define the unknown. Let's call x the number of liters of 12% acid solution to be used.
Since we're going to mix the solutions, our new solution will have (x+5) liters.
How much acid is present in the new solution? Well, the original 5 liters contribute...
Nothing wrong with your answer, you can write the terms in any order:
-5a - 12 is the same as -12 -5a
By the way, the fourth line has a (typing?) mistake, it's missing a negative sign at the beginning, it should read:
-3*4 + -4a +-1a