Question

The daily demand for ice cream cones at a price of $1.20 per cone is 50 cones. At a price of $2.20 per cone, the demand is 30 cones. Use linear interpolation to estimate the demand at a price of $1.50 per cone.

Answer 1

Answer: Linear interpolation is y(x)=-20x+74 and number of cones will be 44

Explanation: (a) Let the linear interpolation is of the form of

y(x)=ax+b ---------------------- (1)

where, x represents the price/cone, y(x) represents total number of cones a price of $x per cone and a and b are the constants to be determined.

The conditions are:

(i) at x=1.20, y(x)=50  

So, from eqn 1

50=a*1.20+b

OR 1.2a+b=50 --------------------------------------------------- (2)

(ii) at x=2.20, y(x)=30. So, from eqn 1

30=2.20*a+b

OR 2.2a+b=30 --------------------------------------------------- (3)

Now, subtracting eqn 3 from eqn 2. we get,

2.2a-1.2a=30-50

a=-20 --------------------------------- (4)

Now, substituting a=-20 into eqn 3, we get,

2.2*(-20)+b=30

-44+b=30

b=30+44

b=74 ---------------------------------------------------- (5)

Now, using eqn 1, 4 and 5, we have,

y(x)=-20x+74 ---------------------------------------------- (6)

This is the required linear interpolation.

(b) Now, for x=$1.50, using eqn 6.

y(1.50)=-20*1.50+74

y(1.50)=-30+74

y(1.50)=44

So, the demond of cone at a price of $1.50 per cone will be 44 cones.