Final answer:
The linear function is y = 3x + 5. The slope represents the rate of change of the number of butterflies with respect to time. The number of butterflies 21 minutes after sunrise would be 68.
Step-by-step explanation:
a. To write the linear function, we need to determine the slope (m) and the y-intercept (b) of the equation. We can use the formula for the equation of a line, y = mx + b, where y represents the number of butterflies and x represents the time after sunrise. We can find the slope by using the formula m = (change in y) / (change in x). In this case, the change in y is 26 - 17 = 9 and the change in x is 7 - 4 = 3. Therefore, the slope is 9/3 = 3. To find the y-intercept, we can substitute one of the points into the equation and solve for b. Let's use the point (4, 17). Plugging in the values, we get 17 = 3(4) + b. Solving for b, we get b = 17 - 12 = 5. So the linear function is y = 3x + 5.
b. The slope of the linear function represents the rate of change of the number of butterflies with respect to time. In this case, the slope of 3 means that for every 1 minute increase in time after sunrise, the number of butterflies increases by 3.
c. To calculate the number of butterflies 21 minutes after sunrise, we can substitute the value of x into the linear function. Plugging in x = 21 into the equation y = 3x + 5, we get y = 3(21) + 5 = 63 + 5 = 68. Therefore, there would be 68 butterflies 21 minutes after sunrise.