Final answer:
To find the expected number of kernels to pop at the optimal temperature, first calculate the vertex of the quadratic equation representing the percentage of kernels that pop as temperature changes. Then multiply the total number of kernels by the percentage at the optimal temperature.
Step-by-step explanation:
The Crazy Carmel Corn company has determined the relationship between the temperature of the oil and the percentage of kernels that pop using a quadratic equation. To calculate the optimal number of popcorn kernels that pop, we must first determine the temperature at which the maximum percentage of kernels pop, as described by the equation P = -1/250t² + 2.8t - 394. Here, P represents the percentage of kernels that pop and t represents the temperature.
The optimal temperature for popping kernels is found by taking the derivative of the equation and setting it equal to zero to find the vertex of the parabola. Once we have the optimal temperature, we substitute it back into the original equation to find the maximum percentage of kernels that pop. Since a typical batch consists of 800 kernels, we'll calculate the expected number of popped kernels at the optimal temperature by multiplying the total number of kernels by the maximum percentage P.
To do this calculation without the derivative, we look for the vertex of the equation, which lies on the line of symmetry of the parabola. The line of symmetry can be found using the equation t = -b/(2a), where a and b are the coefficients from the original equation. Thus, we calculate t = -2.8/(2*(-1/250)). After finding the optimal temperature t, we plug the value back into the percentage equation to find P, and then compute the expected number of kernels: Expected Kernels = P * 800.