Case study (4 Marks)

Question 14 Marks

The feasible region for an L.P.P is shown in the figure given below:

i. What is equation of the line AD ? )
ii. What is the equation of the line BC ?
iii. What are the coordinates of the points B and C?
OR
What are the constraints for the L.P.P.?

Answer

i. From the given figure, $OA =25$ and $OD =50$
The equation of the line AD is $\frac{x}{25}+\frac{y}{50}=1$...(intercept form)
i.e. $2 x+y=50$
ii. The equation of the line BC is $\frac{x}{40}+\frac{y}{20}=1$ i.e. $x +2 y =40$
iii. Solving equations $2 x+y=50$ and $x+2 y=40$ simultaneously, we get $x=20, y-10$
$\therefore$ The coordinates of point $B$ are $(20,10)$.
From the given figure, the coordinates of point $C$ are $(0,20)$.
OR
As (0, 0) lies in the region $2 x+y \leq 50$ and (0,0) also lies in the region $x+2 y \leq 40$,therefore, the constraints for the L.P.P. are $2 x+y \leq 50, x+2 y \leq 40, x \geq 0, y \geq 0$

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Question 24 Marks

The feasible region for an L.P.P. is shown in the adjoining figure. The line CB is parallel to OA.
i. Find the equation of the line OA.
ii. Find the equation of the line BC.
iii. Find the constraints for the L.P.P
OR
Find the minimum value of the objective function Z = 3x - 4y.

Answer

i. $O (0,0), A (6,12)$, so equation of OA is
$
y-0=\frac{12-0}{6-0}(x-0) \Rightarrow y-2 x=0
$
ii. Since $B C$ is parallel to $O A$, so slope of $B C=$ slope of $O A$.
$\therefore$ Equation of BC is $Y -4=2( x -0) \Rightarrow y -2 x =4$.
iii. Constraints for the L.P.P. are
$
y \geq 2 x, y-2 x \leq 4, x \leq 6, x \geq 0, y \geq 0
$
OR
The corner points of the feasible region are (0, 0), (6, 12), (6, 16) and (0, 4). The values of Z = 3x - 4y at these corner points are 0, -30, - 46 and -16. Hence, the minimum value of Z = - 46.

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Question 34 Marks

Loans are an integral part of our lives today. We take loans for a specific purpose - for buying a home, or a car,
or sending kids abroad for education - loans help us achieve some significant life goals. That said, when we talk
about loans, the word “EMI", eventually crops up because the amount we borrow has to be returned to the lender
with interest.
Suppose a person borrows ₹1 lakh for one year at the fixed rate of 9.5 percent per annum with a monthly rest. In
this case, the EMI for the borrower for 12 months works out to approximately ₹8,768.
Example:
In year 2000, Mr. Tanwar took a home loan of ₹3000000 from State Bank of India at 7.5% p.a. compounded
monthly for 20 years.
(a) Find the equated monthly installment paid by Mr. Tanwar.
(b) Find interest paid by Mr. Tanwar in 150th payment.
(c) Find Principal paid by Mr. Tanwar in 150th payment.
OR
Find principal outstanding at the beginning of 193th month.

Answer

Loans are an integral part of our lives today. We take loans for a specific purpose - for buying a home, or a car, or sending kids abroad for education - loans help us achieve some significant life goals. That said, when we talk about loans, the word "EMI", eventually crops up because the amount we borrow has to be returned to the lender with interest.
Suppose a person borrows ₹ 1 lakh for one year at the fixed rate of 9.5 percent per annum with a monthly rest. In this case, the EMI for the borrower for 12 months works out to approximately ₹ 8,768 .
Example:
In year 2000, Mr. Tanwar took a home loan of ₹3000000 from State Bank of India at $7.5 \%$ p.a. compounded monthly for 20 years.
(i) ₹ 24167.82
(ii) ₹ 10458.69
(iii)₹ 13709.13
OR
₹ 410293.41

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Question 44 Marks

Yash wants to prepare a handmade gift box for his friend's birthday at his home. For making lower part of the box, he took a square piece of paper of each side equal to 10 cm .

(a) If x cm be the size of the square piece cut from each corner of the paper of size 10 cm , then a possible value of x will be given by interval:
(b) Volume of the open box formed by folding up the cutting corner can be expressed as:
(c) Find the value of x for which $\frac{d V}{d x}=0$ ?
OR
Yash is interested to maximise the volume of the box, So what will be the side of the square to be cut to maximise the volume?

Answer

Yash wants to prepare a handmade gift box for his friend’s birthday at his home. For making lower part of the box, he took a square piece of paper of each side equal to 10 cm

(i) $(0,10)$
(ii) $V=x(10-2 x)(10-2 x)$
(iii) $\frac{5}{3}, 5$
OR
$\frac{5}{3} cm$

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Applied Maths Case study (4 Marks)

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