Day 3 after-lunch quiz:
1. List the 10 smallest numbers that can be written as a sum of two
perfect squares.
2. List all the distances that can be measured corner-to-corner on a 5 x 5
grid of tiles (if you can't list them all, list as many as you can; if you think
you've got them all, explain just why you think so in a way that will convince
us).
3. A square pipe has cross-sectional dimension 2 feet by 2 feet.
Water moves through the pipe and out the end, moving through the pipe at 5 feet
per second.
- How much water exits the pipe every second?
- If the flow rate is changed so that 50 cubic feet flow out every second,
how fast will water now be moving through the pipe?
- If that pipe carries water from a rectangular reservoir 60 feet by 100
feet by 5 feet deep, how fast is the water level in that reservoir falling?
4. Sketch a graph of y = x^2 + 1, on the interval from x = 0 to x = 3.
Start by making a table of y vs. x, where x takes values 0, 1, 2, 3. Plot these
four points, then connect these points with straight line segments.
- Sketch the vertical line from the x axis to each of the corresponding
points. Your graph will at this point consist of three trapezoids.
- What are the areas of your three trapezoids?
- What is the slope of each of the three line segments at the 'tops' of
your three trapezoids?
- Do the slopes of this graph change at a constant rate?
- By how much does the slope change as you go from the first trapezoid to
the second?
- By how much does the slope change as you go from the second trapezoid to
the third?
- What is the total area of trapezoids 1 and 2?
- What is the total area of trapezoids 1, 2 and 3?