Question #95810
At the beginning of the first day (day 1) after grape harvesting is completed, a grape
grower has 8000 kg of grapes in storage. At the end of day n, for n = 1, 2, . . . , the grape grower
sells 250n/(n + 1) kg of their stored grapes at the local market at the price of $1.50 per kg.
During each day the stored grapes dry out a little so that their weight decreases by 2%.
Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1.
(a) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)
(b) Find the value of wn for n = 1, 2, 3.
(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of
day 1 up to the beginning of day n for n ≥ 1.
Write a MATLAB program to compute wn and rn for n = 1, 2, . . . , num where num is entered
by the user, and display the values in three columns: n, wn, rn with appropriate headings.
Run the program for num = 20. (Use format bank.)
grower has 8000 kg of grapes in storage. At the end of day n, for n = 1, 2, . . . , the grape grower
sells 250n/(n + 1) kg of their stored grapes at the local market at the price of $1.50 per kg.
During each day the stored grapes dry out a little so that their weight decreases by 2%.
Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1.
(a) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)
(b) Find the value of wn for n = 1, 2, 3.
(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of
day 1 up to the beginning of day n for n ≥ 1.
Write a MATLAB program to compute wn and rn for n = 1, 2, . . . , num where num is entered
by the user, and display the values in three columns: n, wn, rn with appropriate headings.
Run the program for num = 20. (Use format bank.)
Expert's answer
(a) w1 = 8000, wn = 0.98*w(n-1) - 250n/(n+1)
(b) w1 = 8000
w2 = 0.98*8000 - 125 = 7715
w3 = 0.98*7340 - 250*2/3 = 7394.0333
(c)
format bank
num = input('Input x >> ');
w = 8000;
r = 0;
disp(' n wn in kg rn in $')
for n = 1:num
if w<0
r = r - 1.5*w; % back does not exist kg to make daily sells
w = 0;
break;
end;
disp([n w r])
w = w*0.98 - 250*n/(n+1);
r = r + 1.5*250*n/(n+1);
end;Output
Input x >> 20
n wn in kg rn in $
1.00 8000.00 0
2.00 7715.00 187.50
3.00 7394.03 437.50
4.00 7058.65 718.75
5.00 6717.48 1018.75
6.00 6374.80 1331.25
7.00 6033.02 1652.68
8.00 5693.60 1980.80
9.00 5357.51 2314.14
10.00 5025.36 2651.64
11.00 4697.58 2992.55
12.00 4374.46 3336.30
13.00 4056.20 3682.45
14.00 3742.94 4030.66
15.00 3434.74 4380.66
16.00 3131.67 4732.23
17.00 2833.75 5085.17
18.00 2540.96 5439.33
19.00 2253.30 5794.60
20.00 1970.73 6150.85
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