Half a Glass Please!
Author: Steve Arnold
Date: 11/12/06

Half a Glass Please!

At a restaurant recently, the waiter offered to fill my glass.

Since I was driving, I asked for just half a glass.

To what height should the glass be filled?

Drag the point marked "a" to explore this problem.

Try and find an algebraic model for the volume of drink in the glass.

Now use the other drag points to try different glasses!

How much is half a glass?

At a restaurant recently, the waiter offered to fill my glass.

Since I was driving, I asked for just half a glass.

To what height should the glass be filled?

Full Glass

Half a Glass

a = 5.2 (slider from 0 to 10, step 0.1)
B = (9, 15)
A = (9, 55)
b = Segment(A, B)
C = Point(b)
n_1 = Segment(C, A)
c = OrthogonalLine(A, b)
G = Point(c)
m_1 = Segment(G, A)
I = Mirror(G, A)
d = OrthogonalLine(C, b)
H = Point(d)
g = Ray(G, H)
s = Segment(C, H)
J = Mirror(H, C)
h = Ray(I, J)
N = Intersect(g, h)
t = Segment(A, N)
a_1 = Segment(N, C)
n = Segment(A, C)
K: C + (0, n_1 a / 10)
k = OrthogonalLine(K, b)
b_1 = Segment(N, K)
q = Segment(A, G)
e = OrthogonalLine(B, b)
D = Point(b)
E = Point(e)
F = Mirror(E, B)
Q, g_1, h_1, j, i = Polygon(G, H, J, I)
L = Intersect(k, g_1)
r = Segment(K, L)
M = Intersect(k, j)
R, l, m, j_1, h_2 = Polygon(L, M, J, H)
p = Segment(C, D)
T3: π r r b_1 / 3 - π a_1 s s / 3 + ""
Glass: π q q t / 3 - π a_1 s s / 3 + ""
X: (a, 0)
Y1: (0, π m_1 m_1 t / 3 - π s s a_1 / 3)
c_1 = OrthogonalLine(Y1, yAxis)
Y2: (0, π m_1 m_1 t / 6 - π a_1 s s / 6)
f_1 = OrthogonalLine(Y2, yAxis)
P: (a, π r r b_1 / 3 - π s s a_1 / 3)
S, f, d_1, e_1 = Polygon(D, E, F)
O = (1, 95)
T = (9, 95)
U = (9, 65)
V = (1, 65)
W, o, t_1, u, v = Polygon(O, T, U, V)
i_1 = Segment(P, X)
k_1 = OrthogonalLine(P, i_1)
Y = Intersect(k_1, yAxis)
l_1 = Segment(P, Y)
p_1 = Segment(C, K)