Math / Calculus

QuestionA cordless leaf blower has a price-demand equation given by $p=D(x)=3375-2.7 x^{2}$ dollars, which gives the price per leaf blower when $x$ leaf blowers are demanded. The price-supply equation for the leaf blower is given by $p=S(x)=1.05 x^{2}$ dollars, which gives the price per leaf blower when $x$ leaf blowers are supplied. Find the consumers' surplus and the producers' surplus.
The consumers' surplus is $\square$ . M̄our answer must begin with \$.)
The producers' surplus is $\square$ (Your answer must begin with \$.)

Studdy Solution

STEP 1

What is this asking? We need to find the extra value consumers get (consumers' surplus) and the extra profit producers get (producers' surplus) when the market for leaf blowers is at equilibrium. Watch out! Don't mix up the supply and demand equations!
Also, remember that surplus is calculated using the equilibrium point, not just any random point.

STEP 2

1. Find the Equilibrium Point
2. Calculate Consumers' Surplus
3. Calculate Producers' Surplus

STEP 3

To find the equilibrium point, we set the demand equal to the supply: $D(x) = S(x)$ .
This tells us when the price consumers are willing to pay equals the price producers are willing to accept.

STEP 4

Let's set the equations equal to each other: $3375 - 2.7x^2 = 1.05x^2$

STEP 5

Now, we solve for $x$ .
First, add $2.7x^2$ to both sides: $3375 = 3.75x^2$

STEP 6

Divide both sides by $3.75$ : $x^2 = \frac{3375}{3.75} = 900$

STEP 7

Take the square root of both sides to find the equilibrium quantity: $x = \sqrt{900} = 30$ So, the equilibrium quantity is $x = \mathbf{30}$ .

STEP 8

Now, plug this value back into either the supply or demand equation to find the equilibrium price.
Let's use the supply equation because it looks simpler: $p = S(30) = 1.05 \cdot (30)^2 = 1.05 \cdot 900 = 945$ So, the equilibrium price is $p = \mathbf{945}$ .
Our equilibrium point is $(30, 945)$ .
Woohoo!

STEP 9

The consumers' surplus is the difference between what consumers are willing to pay and what they actually pay, summed up over all the units sold.
It's the area between the demand curve and the horizontal line at the equilibrium price, up to the equilibrium quantity.

STEP 10

We can calculate this using the definite integral: $CS = \int_0^{30} (D(x) - 945) \, dx$ This integral represents the area we just talked about.

STEP 11

Substitute the demand function: $CS = \int_0^{30} (3375 - 2.7x^2 - 945) \, dx = \int_0^{30} (2430 - 2.7x^2) \, dx$

STEP 12

Now, we evaluate the integral: $CS = \left[ 2430x - \frac{2.7x^3}{3} \right]_0^{30} = \left[ 2430x - 0.9x^3 \right]_0^{30}$

STEP 13

Plug in the limits of integration: $CS = (2430 \cdot 30 - 0.9 \cdot (30)^3) - (0) = 72900 - 24300 = 48600$ So, the consumers' surplus is $\$\mathbf{48,600}$ .
Awesome!

STEP 14

The producers' surplus is the difference between what producers receive and the minimum price they're willing to accept, summed up over all units sold.
It's the area between the supply curve and the horizontal line at the equilibrium price, up to the equilibrium quantity.

STEP 15

We can calculate this using the definite integral: $PS = \int_0^{30} (945 - S(x)) \, dx$ This integral represents the area we just talked about.

STEP 16

Substitute the supply function: $PS = \int_0^{30} (945 - 1.05x^2) \, dx$

STEP 17

Now, we evaluate the integral: $PS = \left[ 945x - \frac{1.05x^3}{3} \right]_0^{30} = \left[ 945x - 0.35x^3 \right]_0^{30}$

STEP 18

Plug in the limits of integration: $PS = (945 \cdot 30 - 0.35 \cdot (30)^3) - (0) = 28350 - 9450 = 18900$ So, the producers' surplus is $\$\mathbf{18,900}$ .
Fantastic!

STEP 19

The consumers' surplus is $\$48,600$ . The producers' surplus is $\$18,900$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the Equilibrium Point2. Calculate Consumers' Surplus3. Calculate Producers' Surplus

STEP 3

To find the **equilibrium point**, we set the demand equal to the supply: D(x)=S(x)D(x) = S(x)D(x)=S(x).This tells us when the price consumers are willing to pay equals the price producers are willing to accept.

STEP 4

Let's set the equations equal to each other: 3375−2.7x2=1.05x23375 - 2.7x^2 = 1.05x^23375−2.7x2=1.05x2

STEP 5

Now, we **solve for** xxx.First, add 2.7x22.7x^22.7x2 to both sides: 3375=3.75x23375 = 3.75x^23375=3.75x2

STEP 6

Divide both sides by 3.753.753.75: x2=33753.75=900x^2 = \frac{3375}{3.75} = 900x2=3.753375​=900

STEP 7

Take the square root of both sides to find the **equilibrium quantity**: x=900=30x = \sqrt{900} = 30x=900​=30 So, the equilibrium quantity is x=30x = \mathbf{30}x=30.

STEP 8

STEP 9

The **consumers' surplus** is the difference between what consumers are willing to pay and what they actually pay, summed up over all the units sold.It's the area between the demand curve and the horizontal line at the equilibrium price, up to the equilibrium quantity.

STEP 10

We can calculate this using the definite integral: CS=∫030(D(x)−945) dxCS = \int_0^{30} (D(x) - 945) \, dxCS=∫030​(D(x)−945)dx This integral represents the area we just talked about.

STEP 11

Substitute the demand function: CS=∫030(3375−2.7x2−945) dx=∫030(2430−2.7x2) dxCS = \int_0^{30} (3375 - 2.7x^2 - 945) \, dx = \int_0^{30} (2430 - 2.7x^2) \, dxCS=∫030​(3375−2.7x2−945)dx=∫030​(2430−2.7x2)dx

STEP 12

Now, we **evaluate the integral**: CS=[2430x−2.7x33]030=[2430x−0.9x3]030CS = \left[ 2430x - \frac{2.7x^3}{3} \right]_0^{30} = \left[ 2430x - 0.9x^3 \right]_0^{30}CS=[2430x−32.7x3​]030​=[2430x−0.9x3]030​

STEP 13

Plug in the limits of integration: CS=(2430⋅30−0.9⋅(30)3)−(0)=72900−24300=48600CS = (2430 \cdot 30 - 0.9 \cdot (30)^3) - (0) = 72900 - 24300 = 48600CS=(2430⋅30−0.9⋅(30)3)−(0)=72900−24300=48600 So, the consumers' surplus is $48,600\$\mathbf{48,600}$48,600.Awesome!

STEP 14

The **producers' surplus** is the difference between what producers receive and the minimum price they're willing to accept, summed up over all units sold.It's the area between the supply curve and the horizontal line at the equilibrium price, up to the equilibrium quantity.

STEP 15

We can calculate this using the definite integral: PS=∫030(945−S(x)) dxPS = \int_0^{30} (945 - S(x)) \, dxPS=∫030​(945−S(x))dx This integral represents the area we just talked about.

STEP 16

Substitute the supply function: PS=∫030(945−1.05x2) dxPS = \int_0^{30} (945 - 1.05x^2) \, dxPS=∫030​(945−1.05x2)dx

STEP 17

Now, we **evaluate the integral**: PS=[945x−1.05x33]030=[945x−0.35x3]030PS = \left[ 945x - \frac{1.05x^3}{3} \right]_0^{30} = \left[ 945x - 0.35x^3 \right]_0^{30}PS=[945x−31.05x3​]030​=[945x−0.35x3]030​

STEP 18

Plug in the limits of integration: PS=(945⋅30−0.35⋅(30)3)−(0)=28350−9450=18900PS = (945 \cdot 30 - 0.35 \cdot (30)^3) - (0) = 28350 - 9450 = 18900PS=(945⋅30−0.35⋅(30)3)−(0)=28350−9450=18900 So, the producers' surplus is $18,900\$\mathbf{18,900}$18,900.Fantastic!

STEP 19

The consumers' surplus is $48,600\$48,600$48,600. The producers' surplus is $18,900\$18,900$18,900.

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1. Find the Equilibrium Point
2. Calculate Consumers' Surplus
3. Calculate Producers' Surplus

To find the equilibrium point, we set the demand equal to the supply: $D(x) = S(x)$ .
This tells us when the price consumers are willing to pay equals the price producers are willing to accept.

Let's set the equations equal to each other: $3375 - 2.7x^2 = 1.05x^2$

Now, we solve for $x$ .
First, add $2.7x^2$ to both sides: $3375 = 3.75x^2$

Divide both sides by $3.75$ : $x^2 = \frac{3375}{3.75} = 900$

Take the square root of both sides to find the equilibrium quantity: $x = \sqrt{900} = 30$ So, the equilibrium quantity is $x = \mathbf{30}$ .

The consumers' surplus is the difference between what consumers are willing to pay and what they actually pay, summed up over all the units sold.
It's the area between the demand curve and the horizontal line at the equilibrium price, up to the equilibrium quantity.

We can calculate this using the definite integral: $CS = \int_0^{30} (D(x) - 945) \, dx$ This integral represents the area we just talked about.

Substitute the demand function: $CS = \int_0^{30} (3375 - 2.7x^2 - 945) \, dx = \int_0^{30} (2430 - 2.7x^2) \, dx$

Now, we evaluate the integral: $CS = \left[ 2430x - \frac{2.7x^3}{3} \right]_0^{30} = \left[ 2430x - 0.9x^3 \right]_0^{30}$

Plug in the limits of integration: $CS = (2430 \cdot 30 - 0.9 \cdot (30)^3) - (0) = 72900 - 24300 = 48600$ So, the consumers' surplus is $\$\mathbf{48,600}$ .
Awesome!

The producers' surplus is the difference between what producers receive and the minimum price they're willing to accept, summed up over all units sold.
It's the area between the supply curve and the horizontal line at the equilibrium price, up to the equilibrium quantity.

We can calculate this using the definite integral: $PS = \int_0^{30} (945 - S(x)) \, dx$ This integral represents the area we just talked about.

Substitute the supply function: $PS = \int_0^{30} (945 - 1.05x^2) \, dx$

Now, we evaluate the integral: $PS = \left[ 945x - \frac{1.05x^3}{3} \right]_0^{30} = \left[ 945x - 0.35x^3 \right]_0^{30}$

Plug in the limits of integration: $PS = (945 \cdot 30 - 0.35 \cdot (30)^3) - (0) = 28350 - 9450 = 18900$ So, the producers' surplus is $\$\mathbf{18,900}$ .
Fantastic!

The consumers' surplus is $\$48,600$ . The producers' surplus is $\$18,900$ .