The demand of particular product decreases as a linear function as the price goes up . A company researched that when the product price is $45, the demand for the product is 1000. For every $5 increase in the price of the product , the demand falls by 5. Which of the following correctly respesents the demand, D for the the prodcut as a function of the product price P?

Options

A. `D(p) = 1000- ((-4)/(5))p`
B. `D(p) = 1000 - (p + 45)`
C. `D(p) = (1000)/(45) - (5)/(4)p`
D. `D(p) = 1000 - (5)/(4) ((4p)/(5) - 36)`

Answer

To solve the problem step-by-step, we need to establish a linear relationship between the demand (D) and the price (P) of the product.

### Step 1: Identify the initial conditions
We know from the problem that:
- When the price (P) is $45, the demand (D) is 1000.
- For every $5 increase in price, the demand decreases by 5.

### Step 2: Determine the slope of the demand function
Since the demand decreases linearly with an increase in price, we can find the slope (m) of the demand function. The slope can be calculated as:
\[ m = \frac{\text{Change in Demand}}{\text{Change in Price}} = \frac{-5}{5} = -1 \]

### Step 3: Use the point-slope form of a linear equation
We can use the point-slope form of the equation of a line, which is:
\[ D - D_1 = m(P - P_1) \]
Where:
- \(D_1 = 1000\) (the demand when the price is $45)
- \(P_1 = 45\) (the price at that demand)

Substituting the known values into the equation:
\[ D - 1000 = -1(P - 45) \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ D - 1000 = -P + 45 \]
Adding 1000 to both sides:
\[ D = -P + 45 + 1000 \]
\[ D = -P + 1045 \]

### Step 5: Finalize the demand function
Thus, the demand function D in terms of price P is:
\[ D = -P + 1045 \]

### Step 6: Verify with the given conditions
1. For \(P = 45\):
\[ D = -45 + 1045 = 1000 \quad \text{(Correct)} \]
2. For \(P = 50\):
\[ D = -50 + 1045 = 995 \quad \text{(Correct)} \]

### Conclusion
The demand function that represents the demand D for the product as a function of the product price P is:
\[ D = -P + 1045 \]