Megha wants to prepare a handmade gift box for her friend's birthday at home. For making lower part of box, she takes a square piece of cardboard of side 20 cm.

Based on the above information, answer the following questions.
(i) If x cm be the length of each side of the square cardboard which is to be cut off from corners of the square piece of side 20 cm, then possible value of x will be given by the interval

(ii) Volume of the open box formed by folding up the cutting corner can be expressed as

(iii) The values of x for which \(\begin{equation} \frac{d V}{d x}=0 \end{equation}\) ,are

(iv) Megha is interested in maximising the volume of the box. So, what should be the side of the square to be cut off so that the volume of the box is maximum?

(v) The maximum value of the volume is

Shobhit's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in' figure. He has 200 ft of wire fencing.
 
Based on the above information, answer the following questions.
(i) To construct a garden using 200 ft of fencing, we need to maximise its

(ii) If x denote the length of side of garden perpendicular to brick wall and y denote the length, of side parallel to brick wall, then find the relation representing total amount of fencing wire.

(iii) Area of the garden as a function of x, say A(x), can be represented as

(iv) Maximum value of A(x) occurs at x equals

(v) Maximum area of garden will be

The Government declare that farmers can get Rs.300 per quintal for their onions on 1^st July and after that,the price will be dropped by Rs. 3 per quintal per extra day.
Shyams father has 80 quintal of onions in the field on 1st July and he estimates that crop is increasing at the rate of 1 quintal per day.

Based on the above information, answer the following questions.
(i) If x is the number of days after 1st July, then price and quantity ofonion respectively can be expressed as

(ii) Revenue R as a function of x can be represented as

(iii) Find the number of days after 1stJuly, when Shyams father attain maximum revenue.

(iv) On which day should Shyam's father harvest the onions to maximise his revenue?

(v) Maximum revenue is equal to

An owner of an electric bi~e rental company have determined that if they charge customers Rs. x per day to rent a bike, where 50 Rs. x Rs. 200, then number of bikes (n), they rent per day can be shown by linear function n(x) = 2000 - 10x. If they charge Rs. 50 per day or less, they will rent all their bikes. If they charge Rs. 200 or more per day, they will not rent any bike. Based on the above information, answer the following questions.

Based on the above information, answer the following questions
(i) Total revenue R as a function of x can be represented as

(ii) If R(x) denote the revenue, then maximum value of R(x) occur when x equals

(iii) At x = 260, the revenue collected by the company is

(iv) The number of bikes rented per day, if x = 105 is

(v) Maximum revenue collected by company is

Mr. Sahil is the owner of a high rise residential society having 50 apartments. When he set rent at Rs. 10000/month, all apartments are rented. If he increases rent by Rs. 250/ month, one fewer apartment is rented. The maintenance cost for each occupied unit is Rs. 500/month. Based on the above information answer the following questions.

Based on the above information answer the following questions.
(i) If P is the rent price per apartment and N is the number of rented apartment, then profit is given by

(ii) If x represent the number of apartments which are not rented, then the profit expressed as a function of x is

(iii) If P = 10500, then N =

(iv) If P = 11,000, then the profit is

Western music concert is organised every year in the stadium that can hold 36000 spectators. With ticket price of Rs. 10, the average attendance has been 24000. Some financial expert estimated that price of a ticket should be determined by the function \(p(x)=15-\frac{x}{3000}\), where x is the number of tickets sold.

Based on the above information, answer the following questions.
(i) The revenue, R as a function of xcan be represented as

(ii) The range of x is

(iii) The value of xfor which revenue is maximum, is

(iv) When the revenue is maximum, the price of the ticket is

(v) How any spectators should be present to maximize the revenue?

A tin can manufacturer designs a cylindrical tin can for a company making sanitizer and disinfector. The tin can is made to hold 3 litres of sanitizer or disinfector.

Based on the above in formation, answer the following questions. 
(i) If r cm be the radius and h em be the height of the cylindrical tin can, then the surface area expressed as a function of r as

(ii) The radius that will minimize the cost of the material to manufacture the tin can is

(iii) The height thatt will minimize the cost of the material to manufacture the tin can is

(iv) If the cost of material used to manufacture the tin can is Rs.100/m² and \(\sqrt[3]{\frac{1500}{\pi}} \approx 7.8\) then minimum cost is approximately

(v) To minimize the cost of the material used to manufacture the tin can, we need to minimize the

A poster is to be formed for a company advertisement. The top and bottom margins of poster should be 9 cm and the side margins should be 6 cm. Also, the area for printing the advertisement should be 864 cm²

Based on the above information, answer the following questions.
(i) If a cm be the width and b cm be the height of poster, then the area of poster, expressed in terms of a and b, is given by

(ii) The relation between a and b is given by

(iii) Area of poster in terms of b is given by

(iv) The value of b, so that area of the poster is minimized, is

(v) The value of a, so that area of the poster is minimized, is

Nitin wants to construct a rectangular plastic tank for his house that can hold 80 ft 3 of water. The top of the tank is open. The width of tank will be 5 ft but the length and heights are variables. Building the tank cost Rs.20 per sq. foot for the base and Rs. 10 per square foot for the side.

Based on the above information, answer the following questions.

(i) In order to make a least expensive water tank, Nitin need to minimize its

(ii) Total cost of tank as a function of h can' be' represented as

(iii) Range of h is

(iv) Value of h at which c(h) is minimum, is

(v) The cost ofleast expensive tank is

Shreya got a rectangular parallelopiped shaped box and spherical ball inside it as return gift. Sides of the box are x, 2x, and x/3, while radius of the ball is r.

Based on the above information, answer the following questions.
(i) If S represents the sum of volume of parallelopiped and sphere, then S can be written as

(ii) If sum of the surface areas of box and ball are given to be constant k² , then x is equal to

(iii) The radius of the ball, when S is minimum, is

(iv) Relation between length of the box and radius of the ball can be represented as

(v) Minimum value of S is

A real estate company is going to build a new residential complex. The land they have purchased can hold at most 4500 apartments. Also, if they make x apartments, then the monthly maintenance cost for the whole
complex would be as follows: Fixed cost = Rs. 50,00,000,Variable cost = Rs.(160x - 0.04x²)

Based on the above information, answer the following questions.
(i) The maintenance cost as a function of x will be

(ii) If C(x) denote the maintenance cost function, then maximum value of C(x) occur at x =

(iii) The maximum value of C(x) would be

(iv) The number of apartments, that the complex should have in order to minimize the maintenance cost, is

(v) If the minimum maintenance cost is attain, then the maintenance cost for each apartment would be

Kyra has a rectangular painting canvas a toatl area of 24ft² which include a border of 0.5ft on the left,right and a border 0.75 ft on the bottom,top inside it.

Based on the above information, answer the following questions.
(i) If Kyra wants to paint in the maximum area: then she needs to maximize

(ii) If x is the length of the outer rectangle, then area of inner rectangle in terms of x is

(iii) Find the range of x.

(iv) If area of inner rectangle is m~imum, then x is equal to

(v) If area of inner rectangle is maximum, then length and breadth of this rectangle are respectively

A magazine company in a town has 5000 subscribers on its list and collects fix charges of Rs. 3000 per year from each subscriber. The company proposes to increase the annual charges and it is believed that for every increase of Rs. 1, one subscriber will discontinue service.
 
Based on the above information, answer the following questions.
(i) If x denote the amount of increase in annual charges, then revenue R, as a function of x can be represented as

(ii) If magazine company increases Rs.500 as annual charges, then R is equal to

(iv) What amount of increase in annual charges will bring maximum revenue?

(v) Maximum revenue is equal to

In a street two lamp posts are 600 feet apart. The light intensity at a distance d from the first (stronger) lamp post is \(\frac{1000}{d^{2}}\) the light intensity at distance d from the second (weaker) lamp post is \(\frac{125}{d^{2}}\) (in both cases the light intensity is inversely proportional to the square of the distance to the light source), The combined light intensity is the sum of the two light intensities coming from both lamp posts.

Based on the above information, answer the following questions
(i) If you are in between the lamp posts, at distance x feet from the stronger light, then the formula for the combined light intensity coming from both lamp posts as function of x, is

(ii) The maximum value of x can not be

(iii) The rainimum value of x can not be

(iv) If l(x) denote the combined light intensity, then lex) will be minimum when x =

(v) The darkest spot between the two lights is

An open water tank of aluminium sheet of negligible thickness, with a square base and vertical sides, is to be constructed in a farm for irrigation. It should hold 32000 I of water, that comes out from a tube well.

Based on above information, answer the following questions.
(i) If the length, width and height of the open tank be x, x and y m respectively, then total surface area of tank is

(ii) The relation between x and y is

(iii) The outer surface area of tank will be minimum when depth of tank is equal to

(iv) The cost of material will be least when width of tank is equal to

(v) If cost of aluminium sheet is Rs.360/m², then the minimum cost for the construction of tank will be

A student Arun is running on a playground along the curve given by y = x² + 7. Another student Manita standing at point (3, 7) on playground wants to hit Arun by paper ball when Arun is nearest to Manita .
 
Based on above information, answer the following questions.
(i) Arun's position at any value of x will be

(ii) Distance (say D) between Arun and Manita will be

(iii) For which real value(s) of x, first derivative of D² w.r.t. 'x' will  Vanish?

(iv) Find the position of Arun when Manita will hit the paper hall.

(v) The minimum value of D is

In a society there is a garden in the shape of rectangle inscribed in a circle of radius 10m as shown in given figure.

Based on the above information, answer the following questions.
(i) If 2x and 2y denotes the length and breadth in metres, of the rectangular part, then the relation between the variables is

(ii) The area (A) of green grass, in terms of x, is given by

(iii) The maximum value of A is

(iv) The value oflength of rectangle, when A is maximum, is

(v) The area of gravelling path is

Two multi-storey buildings (represented by AP and BQ) are on opposite side of a 20 m wide road at point A and B respectively. There is a point R on road as shown in figure.

Based on the above information, answer the following questions.
(i) Area of trapezium ABQP is

(ii) The length PQ is

(iii) Let there be a quantity S such that S = Rp² + RQ², then S is given by

(iv) Find the value of x for which value of S is minimum.

(v) For minimum value of S, find the value of PR and RQ

Rohan, a student of class XII, visited his uncle's flat with his father. He observe that the window of the house is in the form of a rectangle surmounted by a semicircular opening having perimeter 10m as shown in the figure.

(i) If x and y represents the length and breadth of the rectangular region, then relation between x and y can be represented as

(ii) The area (A) of the window can be given by

(iii) Rohan is interested in maximizing the area of the whole window, for this to happen, the value of x should be

(iv) Maximum area of the window is

(v) For maximum value of A, the breadth of rectangular part of the window is

An architecture design a auditorium for a school for its cultural activities. The floor of the auditorium is rectangular in shape and has a fixed perimeter P.

Based on the above information, answer the following questions.
(i) If x and y represents the length and breadth of the rectangular region, then relation between the variable is

(ii) The area (A) of the rectangular region, as a function of x, can be expressed as

(iii) School's manager is interested in maximising the area of floor 'A' for this to be happen, the value of x should be

(iv) The value of y, for which the area of floor is maximum is

(v) Maximum area of floor is