The number of products made per hour at a factory can be...

The number of products made per hour at a factory can be modeled by the function $$ P(t)=-8 t^{2}+52 t+150 $$ where $P(t)$ represents the production per hour, $t$ hours after the start of a shift. The number of damaged products produced per hour at the same factory can be modeled by the function $$D(t)=-0.5 t^{2}+3 t+10$$ where $D(t)$ represents the number of damaged products per hour produced, $t$ hours after the start of a shift. a. Find the number of products produced at this factory 3 hours after the start of a shift. b. Find the number of damaged products produced at this factory 3 hours after the start of a shift. c. Find a new function that gives the number of undamaged products per hour produced, $t$ hours after the start of a shift. d. Use your model to determine the number of undamaged products produced per hour 4 hours after the start of a shift.

The number of products made per hour at a factory can be modeled by the function
$$
P(t)=-8 t^{2}+52 t+150
$$
where $P(t)$ represents the production per hour, $t$ hours after the start of a shift. The number of damaged products produced per hour at the same factory can be modeled by the function
$$D(t)=-0.5 t^{2}+3 t+10$$
where $D(t)$ represents the number of damaged products per hour produced, $t$ hours after the start of a shift.
a. Find the number of products produced at this factory 3 hours after the start of a shift.
b. Find the number of damaged products produced at this factory 3 hours after the start of a shift.
c. Find a new function that gives the number of undamaged products per hour produced, $t$ hours after the start of a shift.
d. Use your model to determine the number of undamaged products produced per hour 4 hours after the start of a shift.

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Key Concepts

Evaluating Functions

This concept involves substituting a specific value into a function to compute its output. In contexts where production or damage counts are modeled with polynomial functions, evaluating the function at a given time provides the instantaneous rate or quantity corresponding to that time.

Function Operations and Subtraction

This concept covers the process of combining functions, particularly by subtracting one function from another. In scenarios where total production and a subset (such as damaged products) are modeled separately, subtracting the subset function from the total production function yields a new function that models the remaining quantity (e.g., undamaged products).

Mathematical Modeling

This concept refers to the use of mathematical functions to represent real-world processes. By formulating equations that describe production rates over time, one can analyze different aspects like total output and damage rates, and then predict or optimize outcomes in a practical setting.

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