Which graph can be used to find the solution(s) to 3 – 4x = –3 – x2 + 4x?
2 answers:
1. Consider the functions f(x)=-4x+3 and g(x)=
the graph of f is a line (f is a linear polynomial function) so we can use 2 points to draw it, say (0, 3) and (2, -5)
the graph of g is a parabola (g is a quadratic polynomial function), so we can factorize the expression as follows:
so the roots of g are 1 and 3, the axis of symmetry is the vertical line through (2,0), so the vertex is calculated at x=2, -(2-3)(2-1)=-(-1)(1)=1. Vertex is the point (2, 1)
2. check the picture attached. Clearly there are 2 solutions, at the 2 intersections of the graphs.
3. Algebraically, the solutions are at :
or 
or 
the graph of f is a line (f is a linear polynomial function) so we can use 2 points to draw it, say (0, 3) and (2, -5)
the graph of g is a parabola (g is a quadratic polynomial function), so we can factorize the expression as follows:
so the roots of g are 1 and 3, the axis of symmetry is the vertical line through (2,0), so the vertex is calculated at x=2, -(2-3)(2-1)=-(-1)(1)=1. Vertex is the point (2, 1)
2. check the picture attached. Clearly there are 2 solutions, at the 2 intersections of the graphs.
3. Algebraically, the solutions are at :
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For this case we have the following equation:
To clear w, we must follow the following steps:
1) The value of t multiplied by c pass to divide the other side of the equation:
2) the value of 1000 is passed to multiply to the other side of the equation:
Answer:
The cleared equation for w is:
To clear w, we must follow the following steps:
1) The value of t multiplied by c pass to divide the other side of the equation:
2) the value of 1000 is passed to multiply to the other side of the equation:
Answer:
The cleared equation for w is:
Answer:
E. (54\sqrt{3}-27\pi\)
Step-by-step explanation:
I draw the figure described by the statement down there so you can see easily how to solve it.
Notice 3 things:
- The length of one side of the hexagon is simply 36 divided by 6. = 6 This is beacause, by nature, every single side of the hexagon has the same length.
- The radius of each circle is half the length of one side of the hexagon. This would mean that the radius of each circle is 3.
- The outer circles have 120°/360° of its area enclose. We get this value because by definition, the hexagon can be divided into 6 regular triangles, each one with every angle equal to 60°. The size of the internal angles of the hexagon will be twice this value.
So, in summary, The area enclosed by the circles is expressed like this:
Acircles = 6 * * π * (r)² + π*r² = 3 * π * r² = 27 π
Now all we need is the area of the regular hexagon, which is simply:
Ahexagon = * p * a
Where p is the perimeter, given by the problem, and a is its apothem, the distance between the center of the hexagon and the middle of one of its sides, that is found by multiplying the length of a side of the hexagon times .
Ahexagon = * 36 * 6 *
= 54
.
Then, the area of the shaded area is equal to:
Ashaded = Ahexagon - Acircles = 54 - 27π
