I have four questions in one.
1. A library crew can shelve books in six hours. Another crew can do the job in four hours. If the first crew begins and is joined by the second crew one hour later, how many hours will it take to finish the job?
2. Find a quadratic function f(x)=ax^2 + bx + c whose graph has a maximum value at 25 and x-intercepts -3 and 2.
3. Suppose Z varies directly as X and Y and inversely as W, and Z=8 when X=6, Y=5, and W=10. Find Z when X=3, Y=4, and W=5.
4. Find the equation of the hyberbola with foci (3, -6) and (3,2) and with 6 as the difference of focal radii.
Even if you only know one I sure would appreciate the help. :)
1. 1/6+1/4x=1
1/4x=5/6
x=20/6 hrs= 3 2/6 hrs
Sure! I can help you with all four questions.
1. To solve this problem, you need to consider the rates at which each crew works. The first crew can shelve books in six hours, so their rate of work is 1/6 books per hour. The second crew can shelve books in four hours, so their rate of work is 1/4 books per hour.
Let's assume the total number of books to shelve is "x". In the first hour, the first crew completes 1/6 of the work, leaving 5/6 of the work remaining. After that, both crews work together at a combined rate of 1/6 + 1/4 = 5/12 books per hour.
So, the time it takes to complete the remaining 5/6 of the work is (5/6) / (5/12) = 12/6 = 2 hours.
Therefore, it will take a total of 1 + 2 = 3 hours to finish the job.
2. To find a quadratic function with the given information, we can use the vertex form of a quadratic function: f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Since the graph has a maximum value at 25, the vertex will be in the form (h, 25). Additionally, the x-intercepts are -3 and 2, so the quadratic function should be zero at those points.
Using the fact that the quadratic function is zero when x = -3 and x = 2, we can set up two equations:
a(-3 - h)^2 + 25 = 0 --(1)
a(2 - h)^2 + 25 = 0 --(2)
Expanding the squares and simplifying the equations will give us a system of equations in terms of h and a:
9a + 25 = 0 --(1)
4a - 4ah + 25 = 0 --(2)
Solving this system of equations will give us the values of h and a. Once we have those values, we can substitute them into f(x) = a(x - h)^2 + 25 to find the quadratic function.
3. In a direct variation equation, Z is directly proportional to both X and Y, and inversely proportional to W. This can be represented by the equation Z = k * (X * Y) / W, where k is the constant of variation.
To solve for k, we can substitute the values given into the equation. When Z = 8, X = 6, Y = 5, and W = 10:
8 = k * (6 * 5) / 10
8 = k * 30/10
8 = 3k
Dividing both sides by 3, we get k = 8/3.
Now, we can use this value to find Z when X = 3, Y = 4, and W = 5:
Z = (8/3) * (3 * 4) / 5
Z = (8/3) * 12/5
Z = 96/15
Z = 32/5
Therefore, when X = 3, Y = 4, and W = 5, Z = 32/5.
4. To find the equation of the hyperbola with the given foci and the difference of focal radii, we can use the standard form of a hyperbola equation:
[(x - h)^2 / a^2] - [(y - k)^2 / b^2] = 1
where (h, k) represents the center of the hyperbola, "a" represents the distance from the center to the vertex, and "b" represents the distance from the center to the co-vertex.
Since the foci of the hyperbola are at (3, -6) and (3, 2), the center of the hyperbola is at (3, -2). Also, the difference in focal radii is given as 6, which means 2a = 6, so a = 3.
To find the value of "b", we can use the formula for the distance between the foci, which is given as 2b:
Distance between foci = 2b
6 = 2b
b = 3
Now, we have all the values needed to write the equation of the hyperbola:
[(x - 3)^2 / 3^2] - [(y + 2)^2 / 3^2] = 1
Simplifying this equation gives us the equation of the hyperbola with the given foci and difference of focal radii.
I hope this helps! Let me know if you have any further questions.