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Math  /  Algebra

QuestionAn employee's monthly productivity MM, in number of units produced, is found to be a function of the number tt of years of service. For a certain product, a productivity function is shown below. Find the maximum productivity and the year in which it is achieved. M(t)=−3t2+132t+170,0≤t≤40M(t)=-3 t^{2}+132 t+170,0 \leq t \leq 40
The maximum productivity is achieved in year 22. The maximum productivity is □\square units.

Studdy Solution

STEP 1

1. The productivity function M(t)=−3t2+132t+170 M(t) = -3t^2 + 132t + 170 is a quadratic function.
2. The quadratic function opens downwards because the coefficient of t2 t^2 is negative.
3. The maximum value of a quadratic function ax2+bx+c ax^2 + bx + c occurs at the vertex, which can be found using the formula t=−b2a t = -\frac{b}{2a} .

STEP 2

1. Identify the vertex of the quadratic function to find the year of maximum productivity.
2. Calculate the maximum productivity using the vertex.

STEP 3

Identify the coefficients a a , b b , and c c from the quadratic function:
Given M(t)=−3t2+132t+170 M(t) = -3t^2 + 132t + 170 , we have: - a=−3 a = -3 - b=132 b = 132 - c=170 c = 170

STEP 4

Use the vertex formula t=−b2a t = -\frac{b}{2a} to find the year of maximum productivity:
t=−1322×(−3) t = -\frac{132}{2 \times (-3)} t=−132−6 t = -\frac{132}{-6} t=22 t = 22

STEP 5

Substitute t=22 t = 22 back into the function M(t) M(t) to find the maximum productivity:
M(22)=−3(22)2+132(22)+170 M(22) = -3(22)^2 + 132(22) + 170

STEP 6

Calculate M(22) M(22) :
M(22)=−3(484)+132(22)+170 M(22) = -3(484) + 132(22) + 170 M(22)=−1452+2904+170 M(22) = -1452 + 2904 + 170 M(22)=1622 M(22) = 1622
The maximum productivity is 1622 \boxed{1622} units.

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