Math Snap

STEP 1

1. The demand equation is given by $p = 1400 - 4q$.
2. Elasticity of demand, $\eta$, is defined as $\eta = \frac{p}{q} \cdot \frac{dq}{dp}$.
3. We need to find $\frac{dq}{dp}$ using the given demand equation.
4. We will verify elasticity and total revenue behavior in specified intervals of $q$.

STEP 2

1. Find $\frac{dq}{dp}$.
2. Express $\eta$ in terms of $q$.
3. Verify elasticity and total revenue for $0 < q < 175$.
4. Verify elasticity and total revenue for $175 < q < 350$.

STEP 3

To find $\frac{dq}{dp}$, we first need to express $q$ in terms of $p$ from the demand equation $p = 1400 - 4q$.
Rearrange the equation:
$$4q = 1400 - p$$ $$q = \frac{1400 - p}{4}$$ Now differentiate $q$ with respect to $p$:
$$\frac{dq}{dp} = \frac{d}{dp} \left( \frac{1400 - p}{4} \right) = -\frac{1}{4}$$

STEP 4

Substitute $\frac{dq}{dp} = -\frac{1}{4}$ into the formula for $\eta$:
$$\eta = \frac{p}{q} \cdot \left(-\frac{1}{4}\right)$$ Substitute $p = 1400 - 4q$ into the expression:
$$\eta = \frac{1400 - 4q}{q} \cdot \left(-\frac{1}{4}\right)$$ Simplify:
$$\eta = -\frac{1400 - 4q}{4q} = -\frac{1400}{4q} + 1$$ $$\eta = -\frac{350}{q} + 1$$

STEP 5

Verify elasticity and total revenue for $0 < q < 175$:
For demand to be elastic, $|\eta| > 1$.
$$\left| -\frac{350}{q} + 1 \right| > 1$$ Solve the inequality:
$$\left| -\frac{350}{q} + 1 \right| > 1$$ $$\left| -\frac{350}{q} \right| > 0$$ $$\frac{350}{q} > 2$$ Solve for $q$:
$$350 > 2q$$ $$q < 175$$ This confirms that demand is elastic for $0 < q < 175$.
Total revenue $R = p \cdot q = (1400 - 4q)q$.
Differentiate $R$ with respect to $q$:
$$\frac{dR}{dq} = 1400 - 8q$$ For $0 < q < 175$, $1400 - 8q > 0$, so total revenue is increasing.

Verify elasticity and total revenue for $175 < q < 350$:
For demand to be inelastic, $|\eta| < 1$.
$$\left| -\frac{350}{q} + 1 \right| < 1$$ Solve the inequality:
$$\frac{350}{q} < 2$$ Solve for $q$:
$$350 < 2q$$ $$q > 175$$ This confirms that demand is inelastic for $175 < q < 350$.
For $175 < q < 350$, $1400 - 8q < 0$, so total revenue is decreasing.
The demand is elastic and total revenue is increasing for $0 < q < 175$. The demand is inelastic and total revenue is decreasing for $175 < q < 350$.