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How many L of an 85% alcohol solution must be mixed with 6 L of a 55% alcohol solution to make a 65% solution?
1 answer:
Answer:
3 Liters of the 85% Solution
Step-by-step explanation:
You have to draw your buckets. I can't on computer so imagine the fraction bar is the bottom of the buckets.
Then we multiply the top of the buckets with the bottom:
85x+330=65x+390
Now we simplify and solve
20x=60
x=3
You need 3 liters of the 85% solution!
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The vertex (minimum) of the quadratic ax² +bx +c is located at x=-b/(2a). This means the minimum value of f(x) will be found at x = -3/(2*1) = -1.5.
Since the vertex of the quadratic is less than 0, the maximum value of the quadratic will be found at x=2, the end of the interval farthest from the vertex.
On the given interval, ...
the absolute minimum value of f is f(-1.5) = ln(1.75) ≈ 0.559616
the absolute maximum value of f is f(2) = ln(14) ≈ 2.639057
Since the vertex of the quadratic is less than 0, the maximum value of the quadratic will be found at x=2, the end of the interval farthest from the vertex.
On the given interval, ...
the absolute minimum value of f is f(-1.5) = ln(1.75) ≈ 0.559616
the absolute maximum value of f is f(2) = ln(14) ≈ 2.639057
Answer:
- vertical scaling by a factor of -4
- horizontal translation 5 units left
- vertical translation 11 units up
Step-by-step explanation:
We notice that the multiplier of the squared term in f(x) is 0.5; in g(x), it is -2, so is a factor of -4 times that in f(x).
If we scale f(x) by a factor of -4, we get ...
-4f(x) = -2(x -2)² -12
In order for the squared quantity to be x+3, we have to add 5 to the value that is squared in f(x). That is, x -2 must become x +3. We have to replace x with (x+5) to do that, so ...
(x+5) -2 = x +3
The replacement of x with x+5 amounts to a translation of 5 units to the left.
We note that the added constant after our scaling changes from +3 to -12. Instead, we want it to be -1, so we must add 11 to the scaled function. That translates it upward by 11 units.
The attached graph shows the scaled and translated function g(x):
g(x) = -4f(x +5) +11
