Would You Rather have the revenue from an amusement park Ferris wheel or carousel ride?
I'm a big fan of questions based on the "Would You Rather?" prompt. This weekend, my son and I visited Atlantic Playland for the first time. We bought a book of tickets and started exploring some of the rides. Our favorites were the Ferris wheel and the Nostalgic Carousel. The Ferris wheel costs us 4 tickets per ride and the carousel cost us 3 tickets per ride. If you were an amusement park operator, which ride would you rather have? Why do you think these rides cost a different number of tickets? Both rides lasted about the same amount of time. The carousel holds more people and is much faster to load and unload. Operating a carousel does not take as much training as operating a Ferris wheel. There is a surprising amount of mental math and estimation require to operate a Ferris wheel. Loading and unloading a Ferris wheel takes a bit of time as the weight of the riders has to be balanced. The Ferris wheel appears to be a more popular ride however and often lots of riders while the carousel was never very full (the Ferris wheel can hold up to 24 people in 12, 2-person seats while the Ferris wheel can hold over 30 at a time).
What do you think? If you operated an amusement park, which ride would you want to have?
Update: This question is now on the Would You Rather website!
My wife and I were enjoying a peanut butter cup after dinner a few nights ago. She was eating a standard peanut butter cup (wide and short) and I was eating a miniature peanut butter cup (narrow and tall). I mentioned that the economy rate of mine was higher than hers and therefore my peanut butter cup had a higher peanut butter to chocolate ratio (the economy rate of a container is the ratio of its volume to its surface area. The higher the ratio, the more economical the container is). A 'lively discussion' ensued with a few pauses for the rolling of eyes. Time to break out a ruler, calculator, and the formula for the volume of a frustum (a truncated cone or pyramid) which is the shape of a peanut butter cup.
The volume of a frustum is given by the following formula:
where R and r are the radii of the top and bottom circles of the truncated cone. The formula can be derived without calculus by taking the entire cone and subtracting the tip to make a frustum. When Heron of Alexandria derived the formula for the volume of a square based frustum, he created the the Heronian mean. The Heronian mean of two numbers A and B is the weighted mean of their arithmetic mean and geometric mean. With this, he was one of the first people to encounter imaginary numbers. The formula below is the general formula for a frustum where A and B are the upper and lower base of the frustum:
Lets find the total volume of the big peanut butter cup first. R = 46 mm, r = 38 mm and h = 10 mm. Therefore the volume is 55585.25 mm^3. If we assume that the chocolate is a uniform thickness of 1 mm then we can easily calculate the volume of the peanut butter by subtracting 2 mm from each measurement and recalculating giving 40346.43 mm^3. That means that the volume of chocolate is the difference of these two values or 15238.82 mm^3. This gives us a peanut butter to chocolate ratio of 2.65
Now time to measure the little peanut butter cup. R = 27 mm, r =22 mm and h = 14 mm. Given these measurements, the total volume is 26492.00 mm^3, the volume of peanut butter is 19163.72 mm^3 and the volume of chocolate is 7328.28 mm^3.
This gives us a peanut butter to chocolate ratio for the little peanut butter cup of 2.61. Nearly identical. But the thickness of the chocolate is far from uniform as this cross section shows. The chocolate is actually quite a bit thicker on top and bottom of the little peanut butter cup than on the big one. The peanut butter in the little peanut butter cup actually looks closer to a half-sphere than a frustum. So if you love the chocolate you should eat the little peanut butter cups and if you love the peanut butter, you should eat the big peanut butter cup.
What is the tallest man-made structure around Halifax? Ask students to brainstorm a few ideas. Students might suggest a tall building. Some of the tallest buildings around Halifax are the Maritime Centre 78 m (256 ft), Purdy's Wharf 88 m (289 ft) and Fenwick Tower, the tallest building in Halifax at 98 m (322 ft) tall. Students might also suggest one of the two harbour bridges. The towers on the MacDonald Bridge are 103 m (338 ft) high and the towers on the MacKay Bridge are 96 m (315 ft). The towers on the MacDonald bridge are taller than the tallest buildings in Halifax. An observant student might even suggest the red and white painted smokestacks at the Tufts Cove Generating station. The smokestacks are tall indeed. From the picture below, you can see that the smokestacks are taller than the bridge towers of the MacKay Bridge.
You might ask students how you know by looking at the picture that the smokestacks are taller than the bridge towers. This would be a fun opportunity to talk about perspective.
We can check out the height of the smokestacks by using a little trigonometry.
I found a spot across the harbour from Tuft's Cove to measure the angle to the top of the smokestacks using a clinometer. (My favourite school/education clinometer is the Invicta MK1 Clinometer... not only does it have a cool name, it looks really cool as well!) It was an angle of elevation of 8 degrees. Next I used Google Earth to see that my distance to the centre stack is approx. 1160 metres. So that means tan(8) = x/1160. Solving for x gives us x = 1160*tan(8) = 163 metres. I emailed Emera an they said that the stacks are actually 500 ft. (152.4 m) tall. So I'm about 11 metres off. Not bad considering that at this distance, a variation of 1 degree is about 20 meters. The actual angle should have been about 7.5 degrees vice 8 degrees.
So, how could I minimize the amount of error? As I get farther away, the tangent value gets smaller but the distance that I'm multiplying by gets bigger. At what angle does 1 degree of error create the least amount of difference in the height being measured?
But is this the tallest man-made structure in Halifax? I used to think so, but I was only considering free standing structures. There is a radio transmission tower that is taller but not free standing... the tower has guy wires to hold it up. The CBC radio tower on Geizer’s Hill is even taller than the smokestacks at Tuft's Cove.
So how tall is this tower? I drove up to the top of Geizer's Hill to find out. I found a spot level to the base of the tower at a distance of 475 meters along Washmill Lake Dr. From this spot, my clinometer measured an angle of inclination to the top of the tower of about 23 degrees. 475 x tan(23) = 201 meters (about 659 ft).
A bit of digging led me to a website that stated the antenna height above ground level for the CBC radio tower is 192 m (629 ft). My measurement was only 9 meters different from this height... pretty close. This is so far the tallest thing I've found around Halifax. Let me know if you find something taller!