Math Problem Statement
A supplier supplies 50 football scarves when the price is £6 each and 90 units when the price is £11 each.
(a)Determine the equation of the supply function in the form P = h(Q).
(b)How many additional scarves are supplied for each successive £1 increase in price?
(c)Calculate the quantity supplied when the price is £8.5 per scarf.
(d)Calculate the price when 120 scarves are supplied.
(e)What is the price below which no scarves are supplied?
Solution
We are tasked with finding the supply function , where is the price per scarf and is the quantity of scarves supplied.
Part (a) – Determine the equation of the supply function
We know that the supplier supplies:
- 50 scarves when the price is £6, and
- 90 scarves when the price is £11.
This gives us two points on the price-quantity plane: and .
The supply function is assumed to be linear, so we use the general form of a line equation: Where:
- is the slope (rate of change of price with respect to quantity),
- is the y-intercept (price when ).
Step 1: Find the slope
The slope is the change in price divided by the change in quantity:
Step 2: Find the intercept
We substitute one of the points, say , into the equation :
Thus, the supply function is:
Part (b) – Additional scarves supplied for each £1 increase in price
To find how many additional scarves are supplied for each £1 increase in price, we calculate the inverse of the slope: This means that for each £1 increase in price, 8 additional scarves are supplied.
Part (c) – Calculate the quantity supplied when the price is £8.5
Substitute into the supply function to solve for : Add 0.25 to both sides: Multiply both sides by 8: So, 70 scarves are supplied when the price is £8.5.
Part (d) – Calculate the price when 120 scarves are supplied
Substitute into the supply function to solve for : Thus, the price is £14.75 when 120 scarves are supplied.
Part (e) – The price below which no scarves are supplied
To find the price below which no scarves are supplied, set in the supply function: This implies that if the price is below £0.25, no scarves will be supplied.
Would you like more details on any part? Here are 5 related questions to deepen your understanding:
- How is the supply function different from a demand function in economics?
- Why do we assume the supply function is linear in this case?
- How would the supply curve change if the supplier increased prices more steeply?
- Can you explain the significance of the y-intercept in the supply function?
- What are the practical limitations of a linear supply model in real-world scenarios?
Tip: When dealing with linear equations, the slope represents the rate of change, and the y-intercept tells you the starting point.
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Supply and Demand
Algebra
Formulas
Linear equation P = mQ + c
Slope formula m = (P2 - P1) / (Q2 - Q1)
Theorems
Linear relationship in supply and demand
Suitable Grade Level
Grades 10-12
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