# I made a ascii text file (hw_Q_dat.txt) and cut/paste the data into it, 
# including a line of header. Then, 

dat = read.table("hw_Q_dat.txt", header=T)

x1 = dat[,1]
x2 = dat[,2]
y = dat[,3]

# a)

lm.1 = lm(y ~ x1 + x2 + I(x1^2) + I(x2^2) + I(x1*x2))
lm.1

# (Intercept)           x1           x2      I(x1^2)      I(x2^2)   I(x1 * x2)  
#    51.19338     -2.14571     -1.87120      0.06305      0.02733      0.13171  

# b) 

summary(lm.1)

# Multiple R-squared: 0.4078,
# This means that about 41% of the variance in Strength can be attributed
# to (or explained by) Depth and Water Content through the expression
# given above in lm().

# c) 

lm.2 = lm(y ~ x1 + x2 )
lm.2

# (Intercept)           x1           x2  
#    38.3508      -0.2588      -0.3015  

# d) 

summary(lm.2)

# Multiple R-squared: 0.3648.
# About 36% of the variance in Strength can be attributed to (or explained by) 
# Depth and and Water Content through the linear expression in lm() .

# e)

# Given that the R2 increases from 36% to 41%, it's reasonable to conclude
# that at least one of the higher-order terms provides useful information
# about Strength.