Sure! Let's solve the problems step-by-step.
### 22. Find the vertex and axis of symmetry
We are given the quadratic function:
[tex]\[ y = 0.5x^2 - 12x + 150 \][/tex]
This equation is in the standard form of a quadratic function:
[tex]\[ y = ax^2 + bx + c \][/tex]
where [tex]\( a = 0.5 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 150 \)[/tex].
The vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substituting [tex]\( a = 0.5 \)[/tex] and [tex]\( b = -12 \)[/tex]:
[tex]\[ x = -\frac{-12}{2 \cdot 0.5} = \frac{12}{1} = 12 \][/tex]
So, the x-coordinate of the vertex is 12. To find the y-coordinate of the vertex, substitute [tex]\( x = 12 \)[/tex] back into the original function:
[tex]\[ y = 0.5(12)^2 - 12(12) + 150 \][/tex]
Calculate:
[tex]\[ y = 0.5 \cdot 144 - 144 + 150 \][/tex]
[tex]\[ y = 72 - 144 + 150 \][/tex]
[tex]\[ y = 78 \][/tex]
So, the vertex of the function is [tex]\((12, 78)\)[/tex].
The axis of symmetry is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
[tex]\[ x = 12 \][/tex]
### 23. What is the minimum cost?
In the context of a quadratic function that opens upwards (as the coefficient of [tex]\( x^2 \)[/tex] is positive), the vertex represents the minimum point of the function. Therefore, the minimum cost is the y-coordinate of the vertex.
From our calculations, the y-coordinate of the vertex is 78. Thus, the minimum cost is:
[tex]\[ 78 \text{ dollars} \][/tex]
### 24. How many cakes should be prepared each month to yield the minimum cost?
To find the number of cakes that should be prepared each month to yield the minimum cost, we look at the x-coordinate of the vertex. From our calculations, the x-coordinate of the vertex is 12.
Therefore, the baker should prepare:
[tex]\[ 12 \text{ cakes each month} \][/tex]
In summary:
- The vertex of the function is at (12, 78).
- The axis of symmetry is [tex]\( x = 12 \)[/tex].
- The minimum cost is 78 dollars.
- The number of cakes to be prepared each month to achieve this minimum cost is 12 cakes.
