A circular oil spill continues to increase in size. The radius of the oil spill, in miles, is given by the function \( r(t)=0.5+2 t \), where \( t \) is the time in hours. The area of the circular region is given by the function \( A(r)=m r^{2} \), where \( r \) is the radius of the circle at time \( t \). Explain how to write a composite function to find the area of the region at time \( t \). DONEIr

To find the area of the oil spill as it changes over time, you need to compose the two functions you've been given. Start with the radius function \( r(t) = 0.5 + 2t \), which tells you how the radius increases as time passes. To find the area at any time \( t \), substitute \( r(t) \) into the area function. This leads to the composite function \( A(r(t)) = m(0.5 + 2t)^{2} \). Now, you have an equation that represents the area of the oil spill in terms of time \( t \)! Now, you can see how quickly this oil spill expands! This functions not only help in conceptual understanding but also influence real-world scenarios like environmental assessments. Determining the area of the oil spill is essential for estimating how much natural resource is affected and for mobilizing the necessary teams for cleanup operations. The more we understand these mathematical relationships, the better equipped we are to tackle environmental challenges effectively!