Joyce is buying a pair of jeans for $39.99. The jeans are on sale at a 20% discount and the sales tax is 7%. Write a function T(x) to represent the total cost of an item with 7% sales tax. Find the composite function (T of S)(x).Use the composition to find the final cost of the jeans.
Jason bought a vase with $50 gift certificate. The vase was marked 33% off, and the sales tax was 5.5%. If he paid $45.95 for the vase, use the composition of functions to determine the original price of the vase.
An auto mechanic charges $35 per hour for labor plus $45 for parts for a tune up on six-cylinder car. The cost of a tune up function determined by the formula C(t)=35t+45. The amount of time for the tune up is determined by a formula given in the manual, A(x)=0.5x, where x is a numerical code for the particle make of car.Write a formula for the cost of tune up in terms of the numerical code. If the numerical code for certain car is 3, determine the final cost of the tune up.
1. To find the function T(x) that represents the total cost of an item with 7% sales tax, we need to add the original price of the item plus the sales tax.
T(x) = x + 7% of x
2. To find the composite function (T of S)(x), we need to apply the function T(x) to the discounted price of the item (after applying the 20% discount).
(T of S)(x) = T(S(x))
= T(0.8x)
Substituting T(x) into the equation:
(T of S)(x) = (0.8x) + 7% of (0.8x)
3. To find the final cost of the jeans, we substitute the original price x = $39.99 into the composite function (T of S)(x).
Final Cost = (T of S)($39.99)
= (0.8 * $39.99) + 7% of (0.8 * $39.99)
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1. To determine the original price of the vase using the composition of functions, we need to find the inverse of the composition function. This will allow us to find the input value that gives us the output of $45.95.
Let the original price of the vase be denoted as P.
(T of S)(P) = $45.95
Substitute the functions:
(0.67P) + 5.5% of (0.67P) = $45.95
Solve for P using the equation above.
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1. To write a formula for the cost of tune-up in terms of the numerical code, we need to substitute the formula A(x) = 0.5x into the cost formula C(t) = 35t + 45.
Cost of tune-up = C(A(x))
= C(0.5x)
= 35(0.5x) + 45
2. To find the final cost of the tune-up for a numerical code of 3, substitute x = 3 into the cost formula.
Final Cost = C(A(3))
= 35(0.5*3) + 45
= 35(1.5) + 45
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Please note that the detailed calculations and solutions depend on the given information and formulas provided.
To find the total cost of an item with 7% sales tax, we need to add the original price of the item to the amount of sales tax.
The formula for calculating the amount of sales tax on an item is: sales tax = (percent tax / 100) * price
So we can write the function T(x) to represent the total cost of an item with 7% sales tax as: T(x) = x + (7/100) * x
Now, we need to find the composite function (T of S)(x) by applying the discount first and then the sales tax.
The formula for calculating the discounted price of an item is: discounted price = original price - (percent discount / 100) * original price
So the discounted price after a 20% discount on an item with original price x is: S(x) = x - (20/100) * x
To find the final cost of the jeans, we can apply the composite function (T of S)(x) as follows:
Final cost = (T of S)(39.99) = T(S(39.99)) = T(39.99 - (20/100) * 39.99) = T(31.992)
Now, substituting the value into the T(x) function, we can find the final cost:
Final cost = 31.992 + (7/100) * 31.992 = 31.992 + 2.23944 = 34.23144
Therefore, the final cost of the jeans is $34.23.
Moving on to the second question about Jason's vase:
We want to find the original price of the vase before the discount. Let's assume the original price is x.
Given that Jason paid $45.95 for the vase after a 33% discount and 5.5% sales tax, we can set up the composite function equation:
(T of S)(x) = 45.95
First, we apply the sales tax to the discounted price:
T(x) = x + (5.5/100) * x = 1.055x
Now, applying the discount to the original price and then adding the sales tax:
(T of S)(x) = T( x - (33/100) * x) = T(0.67x) = 1.055 * 0.67x
Setting this equal to $45.95, we can solve for x:
1.055 * 0.67x = 45.95
0.70985x = 45.95
x ≈ 64.709
Therefore, the original price of the vase was approximately $64.71.
Moving on to the third question about the cost of a tune-up:
We are given the cost of labor per hour as $35 and the cost of parts as $45. The amount of time for the tune-up depends on the numerical code for the car make, denoted as x, and is given by the function A(x) = 0.5x.
To find the cost of the tune-up in terms of the numerical code, we can use the formula for the cost of labor per hour and the cost of parts:
C(t) = 35t + 45
Since the time for the tune-up is determined by the numerical code, we can substitute 0.5x for t:
C(x) = 35(0.5x) + 45
Simplifying this equation, we get:
C(x) = 17.5x + 45
To determine the final cost of the tune-up for a numerical code of 3, we substitute x = 3 into the formula for the cost of the tune-up:
C(3) = 17.5(3) + 45 = 52.5 + 45 = 97.5
Therefore, the final cost of the tune-up for a numerical code of 3 is $97.5.