There are 2 possibilities for where A can be: one where C is 30° (and A is 60°) and another where C is 60° (and A is 30°). Since it's not specified, we can find both.
30°:
drawing the triangle on a graph, you can see that point C is 4 units above point B, so we know that one side of the triangle is 4. Once we find the other "leg" of the triangle (the one that's parallel to the x-axis), we can just add that value to B to find the x coordinate of A.
If angle C is 30°, using the side ratios of a 30-60-90 triangle, that side is "a√3", and the side we're looking for is a. So, to find a, we just divide 4 by √3. In that case, point A is 4/(√3) units to the right of -2√3. We can rationalize 4/(√3) like this:
(4√3)/3
and then add that to 2√3:
(4√3)/3 + -2√3
(4√3)/3 + (-6√3)/3 = (-2√3)/3
We know that the x-coordinate of A is (-2√3)/3, and the y-coordinate is -1 because B is a right angle and we're just moving horizontally. So, if C is 30° and A is 60°, point A is at ((-2√3)/3, -1).
60°:
in this case, the leg we know is "a" and the leg we're looking for is "a√3". So, we can multiply 4 by √3 to get the distance from B:
4 x √3 = 4√3
4√3 + -2√3 = 2√3
So the x-coordinate of A here is 2√3, and the y-coordinate is still -1: (2√3, -1).
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where
.
By continuity of
, you have
As
in the numerator, you approach
; in the denominator, you approach
. So you have an indeterminate form
. Provided the limit indeed exists, L'Hopital's rule can be used.
Now the numerator approaches
, while the denominator approaches
, suggesting the limit above is 2. This means
<span>The total number of hours worked between Ruby, Isaac, and Svetlana, hard-working friends with great names, is 126 and is equal to r + i + s. 126 = r + i + s. The relationships between Svetlana and Ruby's hours can be written as s = 4r. The relationship between Ruby and Isaac's hours can be written as r = i + 6, or i = r - 6. With these equations, we can return to the original equation and substitute for/ replace Svetlana and Isaac's hours so that the whole equation is in terms of r: 126 = (r) + (r-6) + (4r). Simplify this to 126 = 6r - 6 and then again to 132 = 6r and then at last to 22 = r, the number of hours that Ruby worked. Now returning to the other equations, we can plug in that r value to solve for i -- i = 22 - 6, or 16 hours for Isaac -- and for s -- s = 4(22), or a whopping 88 hours for Svetlana. We conclude that Isaac worked for 16 hours, Ruby worked for 22 hours, and Svetlana worked for 88 hours. 88 + 22 + 16 = 126 hours.</span>
Answer:
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Step-by-step explanation:
