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Math  /  Algebra

QuestionFor the demand equation p=1400−4qp=1400-4 q, verify that demand is elastic and total revenue is increasing for 0<q<1750<q<175. Verify that demand is inelastic and total revenue is decreasing for 175<q<350175<q<350.
Begin by finding η\eta in terms of qq. The formula for η\eta is η=pq⋅dqdp\eta=\frac{p}{q} \cdot \frac{d q}{d p}. Since p=1400−4q,dqdp=p=1400-4 q, \frac{d q}{d p}= □\square (Simplify your answer.)

Studdy Solution

STEP 1

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

STEP 2

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

STEP 3

To find dqdp\frac{dq}{dp}, we first need to express q q in terms of p p from the demand equation p=1400−4q p = 1400 - 4q .
Rearrange the equation:
4q=1400−p 4q = 1400 - p q=1400−p4 q = \frac{1400 - p}{4}
Now differentiate q q with respect to p p :
dqdp=ddp(1400−p4)=−14 \frac{dq}{dp} = \frac{d}{dp} \left( \frac{1400 - p}{4} \right) = -\frac{1}{4}

STEP 4

Substitute dqdp=−14\frac{dq}{dp} = -\frac{1}{4} into the formula for η\eta:
η=pq⋅(−14) \eta = \frac{p}{q} \cdot \left(-\frac{1}{4}\right)
Substitute p=1400−4q p = 1400 - 4q into the expression:
η=1400−4qq⋅(−14) \eta = \frac{1400 - 4q}{q} \cdot \left(-\frac{1}{4}\right)
Simplify:
η=−1400−4q4q=−14004q+1 \eta = -\frac{1400 - 4q}{4q} = -\frac{1400}{4q} + 1 η=−350q+1 \eta = -\frac{350}{q} + 1

STEP 5

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

STEP 6

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

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