Problem 1 :
The total cost of airfare on a given route is comprised of the base cost C and the fuel surcharge S in rupee. Both C and S are functions of the mileage m; C(m) = 0.4m + 50 and S(m) = 0.03m. Determine a function for the total cost of a ticket in terms of the mileage and find the airfare for flying 1600 miles.
Solution :
Total cost = C(m) + S(m)
= 0.4m + 50 + 0.03m
T(m) = 0.43 m + 50
To find the airfare for flying 1600 miles, we have to apply 1600 instead of m.
T(m) = 0.43 (1600) + 50
= 688 + 50
T(m) = 738
So, the total cost of airfare for flying 1600 miles is 738.
Problem 2 :
A salesperson whose annual earnings can be represented by the function A(x) = 30, 000 + 0.04x, where x is the rupee value of the merchandise he sells. His son is also in sales and his earnings are represented by the function S(x) = 25, 000 + 0.05x. Find (A + S)(x) and determine the total family income if they each sell Rupees 1, 50, 00, 000 worth of merchandise.
Solution :
(A + S)(x) = A(x) + S(x)
A(x) = 30, 000 + 0.04x -------(1)
S(x) = 25, 000 + 0.05x -------(2)
(1) + (2)
= 30, 000 + 0.04x + 25, 000 + 0.05x
A(x) + S(x) = 55000 + 0.09x
Here x = 15000000
A(x) + S(x) = 55000 + 0.09(15000000)
= 55000 + 1350000
Total income = 1405000
So, the required income is 1405000.
Problem 3 :
The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day the function for exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee.
Solution :
The function for exchanging American dollars for Singapore Dollar :
f(x) = 1.23x
S.D = 1.23 (A.D)
Here "x" stands for American dollar and f(x) stands for Singapore dollar.
A.D = S.D/1.23 ----(1)
Exchanging Singapore Dollar to Indian Rupee is
g(y) = 50.50y
I.R = 50.50 (S.D)
S.D = I.R/50.50
Applying S.D = I.R/50.50 in(1), we get
A.D = (I.R/50.50)/1.23
A.D = I.R/62.115
I.R = 62.115 AD
So, the exchange rate of American dollars in terms of Indian rupee is I.R = 62.115 AD.
Problem 4 :
The owner of a small restaurant can prepare a particular meal at a cost of Rupees 100. He estimates that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200−x. Express his day revenue, total cost and profit on this meal as functions of x.
Solution :
Cost price of the meal = 100
Selling price = x
Number of customers = 200 - x
1 day revenue = No of customers ⋅ x
= (200 - x) ⋅ x
1 day revenue = 200 x - x^{2}
Total cost = Cost of meal ⋅ No of customers
= 100 ⋅ (200 - x)
= 20000 - 100x
Profit = Total cost - 1 day revenue
(200 - x) ⋅ x - 100 ⋅ (200 - x)
Problem 5 :
The formula for converting from Fahrenheit to Celsius temperatures is y = (5x/9) − (160/9). Find the inverse of this function and determine whether the inverse is also a function
Solution :
y = (5x - 160)/9
9y = 5x - 160
5x = 9y + 160
x = (1/5) (9y + 160)
f^{-1} (x) = (1/5) (9x + 160)
f^{-1} (x) = (9x/5) + (160/5)
f^{-1} (x) = (9x/5) + 32
Inverse is also a function.
Problem 6 :
A simple cipher takes a number and codes it, using the function f(x) = 3x−4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x (by drawing the lines).
Solution :
f(x) = 3x−4
Let y = 3x - 4
3x = y + 4
x = (y + 4)/3
f^{-1}(x) = (x + 4)/3
Inverse is also a function.
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