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Answer :
To find the maximum value of the function f(x) = -1.8x² + 16.12x - 38.5, determine the vertex of the parabola, which is done by using the formula h = -b/(2a) to find the x-coordinate of the vertex, and then plug the value of h back into the function to find the y-coordinate, k, which is the maximum value. The calculation results in an approximate maximum value of 29.89 at x = 4.47778.
To find the maximum value of the function f(x) = -1.8x² + 16.12x - 38.5, we can use the vertex form of a quadratic equation. The vertex form is given by f(x) = a(x - h)² + k, where (h,k) is the vertex of the parabola. Because the coefficient of the x² term is negative (-1.8), we know that the parabola opens downward, and thus the vertex represents the maximum point on the graph.
To find the vertex, we use the formula h = -b/(2a), where a is the coefficient of x² and b is the coefficient of x. For our function, a = -1.8 and b = 16.12, so h = -16.12/(2 * -1.8) = 4.47778. Plugging h back into the function to find k, we get k = -1.8(4.47778)² + 16.12(4.47778) - 38.5. Calculating k will give us the maximum value of the function.
After computing, we find that the maximum value of f(x) is at x = 4.47778 and the maximum value, or k, is approximately 29.89 when rounded to the nearest hundredth.
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Rewritten by : Barada
The maximum value of [tex]\( f(x) = -1.8x^2 + 16.12x - 38.5 \)[/tex] is approximately -2.34 at [tex]\( x \approx 4.47 \)[/tex].
To find the maximum value of the function [tex]\( f(x) = -1.8x^2 + 16.12x - 38.5 \)[/tex], we can use the vertex formula. The vertex of a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex] is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex] .
1. First, identify the coefficients:
- [tex]\( a = -1.8 \)[/tex]
- [tex]\( b = 16.12 \)[/tex]
- [tex]\( c = -38.5 \)[/tex]
2. Next, plug the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the vertex formula to find the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
[tex]\[ x = -\frac{16.12}{2*(-1.8)} \][/tex]
[tex]\[ x = -\frac{16.12}{-3.6} \][/tex]
[tex]\[ x \approx 4.47 \][/tex]
3. Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging [tex]\( x \)[/tex] back into the original function [tex]\( f(x) \)[/tex] :
[tex]\[ f(4.47) = -1.8(4.47)^2 + 16.12(4.47) - 38.5 \][/tex]
[tex]\[ f(4.47) \approx -1.8(19.9609) + 16.12(4.47) - 38.5 \][/tex]
[tex]\[ f(4.47) \approx -35.92962 + 72.0924 - 38.5 \][/tex]
[tex]\[ f(4.47) \approx -2.33722 \][/tex]
So, the maximum value of the function is approximately [tex]\( -2.34 \)[/tex] when [tex]\( x \)[/tex] is around [tex]\( 4.47 \)[/tex] .
