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?



We are tasked with finding the supply function \( P = h(Q) \), where \( P \) is the price per scarf and \( Q \) 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: \( (Q_1, P_1) = (50, 6) \) and \( (Q_2, P_2) = (90, 11) \).

The supply function is assumed to be linear, so we use the general form of a line equation:
\[
P = mQ + c
\]
Where:
- \( m \) is the slope (rate of change of price with respect to quantity),
- \( c \) is the y-intercept (price when \( Q = 0 \)).

**Step 1: Find the slope \( m \)**

The slope \( m \) is the change in price divided by the change in quantity:
\[
m = \frac{P_2 - P_1}{Q_2 - Q_1} = \frac{11 - 6}{90 - 50} = \frac{5}{40} = \frac{1}{8}
\]

**Step 2: Find the intercept \( c \)**

We substitute one of the points, say \( (50, 6) \), into the equation \( P = mQ + c \):
\[
6 = \frac{1}{8} \times 50 + c
\]
\[
6 = \frac{50}{8} + c
\]
\[
6 = 6.25 + c
\]
\[
c = 6 - 6.25 = -0.25
\]

Thus, the supply function is:
\[
P = \frac{1}{8} Q - 0.25
\]

### 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:
\[
\frac{1}{m} = \frac{1}{\frac{1}{8}} = 8
\]
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 \( P = 8.5 \) into the supply function \( P = \frac{1}{8} Q - 0.25 \) to solve for \( Q \):
\[
8.5 = \frac{1}{8} Q - 0.25
\]
Add 0.25 to both sides:
\[
8.75 = \frac{1}{8} Q
\]
Multiply both sides by 8:
\[
Q = 8.75 \times 8 = 70
\]
So, 70 scarves are supplied when the price is £8.5.

### Part (d) – Calculate the price when 120 scarves are supplied

Substitute \( Q = 120 \) into the supply function to solve for \( P \):
\[
P = \frac{1}{8} \times 120 - 0.25
\]
\[
P = 15 - 0.25 = 14.75
\]
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 \( Q = 0 \) in the supply function:
\[
P = \frac{1}{8} \times 0 - 0.25 = -0.25
\]
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:

1. How is the supply function different from a demand function in economics?
2. Why do we assume the supply function is linear in this case?
3. How would the supply curve change if the supplier increased prices more steeply?
4. Can you explain the significance of the y-intercept in the supply function?
5. 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.

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?

Explore the solution to a supplier's problem of football scarves pricing and supply. Learn how to find the linear supply function, calculate the quantity supplied at specific prices, and determine how the number of scarves changes with each price increase. Suitable for high school students, Grades 10-12.

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