How many friends are sharing the berries?
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I find it convenient to look at the differences and the rate at which those differences are made up.
8. Jim is closing the $150 gap at the rate of $7.50 per week. He will catch up in
... 150/(7.5/week) = 20 weeks
9. At noon, the price of Stock A has increased by 0.05×3 = 0.15, so is now $15.90, which is $0.63 more than Stock B at that time. The prices are closing the gap at the rate of $0.05 +0.13 = $0.18 per hour, so will be the same after
... $0.63/($0.18/hour) = 3.5 hours . . . . after noon, at 3:30 pm
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You can also write and solve equations for the prices of the stocks. Or you can use a graphing calculator to tell you the solution. When equations are involved, I like to solve them the simplest possible way: let technology do it.
You are given the value at a time, and the rate of change of that value, so the equations are easily written in point-slope form. You will note that the common price at 3:30 pm (15.5 hours after midnight) is one that is not a whole number of cents. (That's usually OK for when trading stocks.)
First integral
Since is a linear function, its graph is a line. This means that you can decompose the area under its graph as the sum of a right triangle and a rectangle, as shown in the attached image.
All the dimensions are fairly easy to deduce: we have
So, the area of the triangle is
For the rectangle, we have
So, the total area is
Second integral
We have
So, this function is the upper half of the unit circle (the two halves are the functions )
Since the area of the unit circle is , the value of the integral is
Third integral
It follows the exact same logic as the first one, you only have to adjust the numbers.
