Two delivery services model their charge, in pounds, for a round of \(x\) parcels as \(f(x) = 1.5x + 8\) for Service A and \(g(x) = 2x + 3\) for Service B. ...
Question 1 Report
Two delivery services model their charge, in pounds, for a round of \(x\) parcels as \(f(x) = 1.5x + 8\) for Service A and \(g(x) = 2x + 3\) for Service B.
- Find the number of parcels for which the two services charge the same amount, and find that charge. (3)
- A courier has a budget of \(\pounds 50\) for one round with Service A. Use \(f^{-1}(x)\) to find the maximum whole number of parcels within budget. (2)
- State, with a reason, which service should be used for a round of 25 parcels. (1)
Answer Details
The crossover point between the two services is found by setting their charge functions equal; a budget limit is handled by using the inverse function to convert a maximum spend directly into a maximum number of parcels, and a specific job size is compared by evaluating both functions.
- \[ 1.5x+8=2x+3 \] [1 mark] \[ 0.5x=5 \] so \[ x=10 \text{ parcels} \] [1 mark], with charge \(f(10)=1.5(10)+8=\pounds23\) [1 mark]
- Rearranging \(y=1.5x+8\) to make \(x\) the subject gives \[ f^{-1}(x)=\frac{x-8}{1.5} \] [1 mark]; with a \(\pounds50\) budget: \[ f^{-1}(50)=\frac{42}{1.5}=28 \text{ parcels} \] [1 mark]
- \[ f(25)=1.5(25)+8=\pounds45.50 \], \[ g(25)=2(25)+3=\pounds53 \]; since \(45.50\lt53\), Service A should be used for a round of 25 parcels. [1 mark]
Service A has the smaller rate per parcel (\(\pounds1.50\) against \(\pounds2\)) but the larger fixed charge (\(\pounds8\) against \(\pounds3\)), which is why it only becomes the cheaper option once the round is large enough, at \(10\) parcels or more, for its lower rate to outweigh its higher starting charge.
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