Let N be the number of spoonfuls of Noisies and C the number of spoonfuls of Crispies. The nutrients are represented by t=thiamine, n=niacin and calorific content by c. So in N spoonfuls we have the quantities of the nutrients: 0.1Nt, Nn, 110Nc; in C spoonfuls we have 0.25Ct, 0.25Cn, 120Cc. We need at least 5n, 400c, and 1t in the combined spoonfuls.
a. 5n=Nn+0.25Cn; so N+0.25C>5; similarly, 110N+120C>400 and 0.1N+0.25C>1.
N>5-0.25C. Or, 0.1N>1-0.25C; N>10-2.5C. Or, N>(400-120C)/110; N>(40-12C)/11.
These requirements can be displayed graphically, using N as a vertical axis and C as a horizontal axis. The inequalities appear as regions above the lines: N=5-0.25C; N=10-2.5C; N=40/11-12C/11. Label the lines t, n and c as appropriate. The intersection of n and t lies above c, so the open region above n and t bounded by the intersection of n and t meets all the minimum nutritional requirements, therefore the intersection point gives us the minimum values of N and C, allowing us to calculate the minimum cost. The equations we need are N=5-0.25C and N=10-2.5C. So, 5-0.25C=10-2.5C, 2.25C=5, C=20/9, from which N=40/9 or 2C. The intersection point is (20/9,40/9). If we work in whole spoonfuls, we can make C=3, and N>5-0.75; N>4.25, so N=5.
b. Cost=3.6N+4.2C cents, so putting in C=3 and N=5 we get cost=5*3.6+3*4.2=30.6 cents or approx 31 cents.
The nutritional content is therefore 1.25mg thiamine, 5.75mg niacin and 910 calories.
[At C=20/9 (2.22) and N=40/9 (4.44) spoonfuls, the cost comes to 25.3 cents and the nutritional content would be 1mg thiamine, 5mg niacin and 755.6 calories. At C=2.5 and N=4.5 spoonfuls, the cost is 26.7 cents and the nutritional content 1.075mg thiamine, 5.125mg niacin and 795 calories.]