How much waffle is needed for a fish-shaped ice cream sandwich?

PREPARATION

To start, I measured the fish. The tail is 10cm wide, and I will use one unit on Desmos as one centimeter. The fish is 2cm thick. I will calculate the area of one side of the fish, double it, and add it with the area of the thickness, which gives the surface area.

The yellow parts under the purple curve are actually spaces not taken up by the fish, so I will account that area as the fish mouth, also in yellow.

The bright pink parts above the black curve (on the right) is also not taken up by the fish, so I will use those areas as the left pink part, the fin, which the black curve is unable to account for.

The red parts between the black and blue curve could not be accounted for, so I used the red parts under the blue curve, which I had to account for in the [0,10] interval, to compensate for the missing area.

The blue parts are to be subtracted from the total area. The blue part on top is because of the overlaps between the black and blue curves, and the blue part on the bottom is to get rid of space the fish does not occupy.

I will use n=10 rectangles for all curves.

CALCULATIONS: TAIL

I started off by calculating the tail. The area of the tail would be given as..

Area under the blue curve (1) — area under the red curve — the area between blue (2) & black curve (1)

1) I used the equation y=-0.2(x-5)² + 6.5 in the interval [0, 10] to calculate the area under the blue curve (1)

Rectangles Area: 48 cm² , n->infinity Area: 48.33 cm²

2) I used the equation y=-0.1(x-5)² + 1.2 in the interval [1.536, 8.464] to calculate the area under the red curve.

Rectangles Area: 5.505 cm², n->infinity Area: 5.547 cm²

3) To calculate the area bound between the black curve and the blue curve, I shifted the graph so the intersection started at the y-axis. The area bound will be given by

Area under the blue curve (2) — area under the black curve (1)

3a) I used the equation y=-0.2(x-2.6)² + 3.4 in the interval [0, 5] to calculate the area under the blue curve (2)

Rectangles Area: 14.62 cm², n->infinity Area: 14.91 cm²

3b) I used the equation y=0.1(x-2.5)² + 1.4 in the interval [0, 5] to calculate the area under the black curve (1)

Rectangles Area: 8.065 cm², n->infinity Area: 8.047 cm²

The area bounded by the blue (2) and black curve (1) is

Area under blue curve (2) — area under black curve (1)

Rectangles Area: 14.62–8.065 = 6.555 cm²

n->infinity Area: 14.91–8.047 = 6.863 cm²

TOTAL AREA OF THE TAIL:

Area under blue curve (1) — area under red curve — area between blue (2) & black curve (1)

Rectangles Area: 48–5.527–6.555 = 35.918 cm²

n->infinity Area: 48.33–5.547–6.863 = 35.92 cm²

CALCULATIONS: BODY

I rotated the fish upside down so that it matches the boundaries that I gave it, also so that the area I need is bounded by the graph and the x-axis.

4) I calculated the area with the equation y = -0.1(x-5.6)²+5.7 in the interval [0, 10] to calculate the area under the black curve (2)

Rectangles Area: 48.14 cm², n->infinity Area: 48.31 cm²

CALCULATIONS: HEAD

Similarly, I moved the picture and shifted my graph so I could have the area I need as the bound region between the curve and the x-axis.

5) I calculated the area with the equation y = -0.18(x-5)²+4.3 in the interval [0, 10] to calculate the area under the purple curve.

Rectangles Area: 27.9 cm², n->infinity Area: 28 cm²

CALCULATIONS: THICKNESS

By measuring, I found that the ice cream sandwich was 2.5 centimeters thick, and the perimeter is 40cm.

We can visualize this as a strip, a rectangle with dimensions 2.5 x 40 = 100 cm²

There is no need to perform n->infinity because it is already a rectangle.

From this, we can see that the surface area of the thickness is 100 cm²

CALCULATIONS: TOTAL

Since the surface area of one side of icecream sandwich is the same as the other side, simply double the area of one side to get the surface area of both sides

Rectangles Area: 111.958x2 = 223.916 cm²

n->infinity Area: 112.23x2 = 224.46 cm²

Then we add the surface area of the thickness to the surface area of the fish side.

Rectangles Area: 223.916+100 = 323.916 cm²

n->infinity Area: 224.46+100 = 324.46 cm²

Thus, the amount of waffle needed for a fish icecream sandwich is estimated to be 323.916 cm², and precisely 324.46 cm²

EXPLANATION:

Why there are differences in rectangle-estimated and n->infinity?

• There are differences because of the sizes of the overestimated areas and underestimated areas compared with each other
• This is because we are working with curves; if we were to be working with a linear line, the area of the overestimated and underestimated by the rectangles will be the same.
• If the rectangle method gives a larger number than the n->infinity method, that means the rectangle method overestimated the area, like in 3b. This is because the small areas that are ‘accounted by the rectangles but not the curve’ are larger than the areas that are ‘accounted by the curve but not the rectangle’. The areas accounted by the rectangle can fill up and compensate for the areas accounted in the curve. Although it fills up, it does overflow, goes over by a little bit. That causes an overestimate of the area.
• If the rectangle method gives a smaller number than the n->infinity method, that means the rectangle method underestimated the area, like in 5. This is because the small areas that are “accounted by the rectangles but not the curve’ are less than the areas that are ‘accounted by the curve but not the rectangle’. In this case, the areas accounted by the rectangle cannot completely fill up the areas accounted by the curve. Therefore, the rectangle method results in less area than the n->infinity method, causing an underestimate of the area.

Why did I choose to do the midpoint of the left and right y values instead of just left and right endpoints?

• Using left endpoint or right endpoint will completely underestimate or overestimate an area. There isn’t one of each to compensate for each other.
• For example, if we chose to take the left endpoint, we would be overestimating by accounting areas above the curve.
• If we choose to take the right endpoint, we would underestimate by not accounting areas under the curve.
• If we choose to take the midpoint, there will be both overestimated areas and underestimated areas. In this case, the overestimated areas can account for the underestimated areas, making the calculated area as accurate as possible.