Math / Calculus

QuestionA cordless leaf blower has a price-demand equation given by $p=D(x)=4286.55-1.05 x^{2}$ 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 $p=S(x)=1.5 x^{2}$ 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 $\square$ (Your 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 value 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: $4286.55 - 1.05x^2 = 1.5x^2$ .
This is where the price consumers are willing to pay matches the price producers are willing to accept.

STEP 4

Now, let's solve for $x$ .
Add $1.05x^2$ to both sides of the equation: $4286.55 = 2.55x^2$ .

STEP 5

Divide both sides by 2.55: $x^2 = \frac{4286.55}{2.55} = 1681$ .

STEP 6

Take the square root of both sides to find the equilibrium quantity: $x = \sqrt{1681} = 41$ .
So, the equilibrium quantity is $x = \textbf{41}$ leaf blowers.

STEP 7

Plug $x = \textbf{41}$ back into either the supply or demand equation to find the equilibrium price.
Let's use the supply equation because it looks simpler: $p = 1.5 \cdot (41)^2 = 1.5 \cdot 1681 = \textbf{2521.50}$ .
The equilibrium price is $\$\textbf{2521.50}$ .

STEP 8

The consumers' surplus is the area between the demand curve and the equilibrium price, integrated from 0 to the equilibrium quantity.
Think of it as the total amount consumers saved compared to what they were willing to pay!

STEP 9

The formula for consumers' surplus is: $\int_0^{41} (4286.55 - 1.05x^2) dx - 41 \cdot 2521.50$ .

STEP 10

Evaluate the integral: $\int_0^{41} (4286.55 - 1.05x^2) dx = [4286.55x - \frac{1.05}{3}x^3]_0^{41} = 4286.55(41) - \frac{1.05}{3}(41)^3 = 175748.55 - 24025.15 = 151723.4$ .

STEP 11

Now, subtract the rectangle's area (equilibrium price times equilibrium quantity): $151723.4 - 41 \cdot 2521.50 = 151723.4 - 103381.5 = \textbf{48341.9}$ .
The consumers' surplus is $\$\textbf{48341.9}$ .

STEP 12

The producers' surplus is the area between the equilibrium price and the supply curve, integrated from 0 to the equilibrium quantity.
It's the extra money producers made compared to the minimum they were willing to accept!

STEP 13

The formula for producers' surplus is: $41 \cdot 2521.50 - \int_0^{41} 1.5x^2 dx$ .

STEP 14

Evaluate the integral: $\int_0^{41} 1.5x^2 dx = [\frac{1.5}{3}x^3]_0^{41} = 0.5(41)^3 = 0.5(68921) = 34460.5$ .

STEP 15

Now, subtract from the rectangle's area: $41 \cdot 2521.50 - 34460.5 = 103381.5 - 34460.5 = \textbf{68921}$ .
The producers' surplus is $\$\textbf{68921}$ .

STEP 16

The consumers' surplus is $\$\textbf{48341.90}$ . The producers' surplus is $\$\textbf{68921.00}$ .

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Studdy Solution

STEP 1

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: $4286.55 - 1.05x^2 = 1.5x^2$ .
This is where the price consumers are willing to pay matches the price producers are willing to accept.

STEP 4

Now, let's solve for $x$ .
Add $1.05x^2$ to both sides of the equation: $4286.55 = 2.55x^2$ .

STEP 5

Divide both sides by 2.55: $x^2 = \frac{4286.55}{2.55} = 1681$ .

STEP 6

Take the square root of both sides to find the equilibrium quantity: $x = \sqrt{1681} = 41$ .
So, the equilibrium quantity is $x = \textbf{41}$ leaf blowers.

STEP 7

Plug $x = \textbf{41}$ back into either the supply or demand equation to find the equilibrium price.
Let's use the supply equation because it looks simpler: $p = 1.5 \cdot (41)^2 = 1.5 \cdot 1681 = \textbf{2521.50}$ .
The equilibrium price is $\$\textbf{2521.50}$ .

STEP 8

The consumers' surplus is the area between the demand curve and the equilibrium price, integrated from 0 to the equilibrium quantity.
Think of it as the total amount consumers saved compared to what they were willing to pay!

STEP 9

The formula for consumers' surplus is: $\int_0^{41} (4286.55 - 1.05x^2) dx - 41 \cdot 2521.50$ .

STEP 10

Evaluate the integral: $\int_0^{41} (4286.55 - 1.05x^2) dx = [4286.55x - \frac{1.05}{3}x^3]_0^{41} = 4286.55(41) - \frac{1.05}{3}(41)^3 = 175748.55 - 24025.15 = 151723.4$ .

STEP 11

Now, subtract the rectangle's area (equilibrium price times equilibrium quantity): $151723.4 - 41 \cdot 2521.50 = 151723.4 - 103381.5 = \textbf{48341.9}$ .
The consumers' surplus is $\$\textbf{48341.9}$ .

STEP 12

The producers' surplus is the area between the equilibrium price and the supply curve, integrated from 0 to the equilibrium quantity.
It's the extra money producers made compared to the minimum they were willing to accept!

STEP 13

The formula for producers' surplus is: $41 \cdot 2521.50 - \int_0^{41} 1.5x^2 dx$ .

STEP 14

Evaluate the integral: $\int_0^{41} 1.5x^2 dx = [\frac{1.5}{3}x^3]_0^{41} = 0.5(41)^3 = 0.5(68921) = 34460.5$ .

STEP 15

Now, subtract from the rectangle's area: $41 \cdot 2521.50 - 34460.5 = 103381.5 - 34460.5 = \textbf{68921}$ .
The producers' surplus is $\$\textbf{68921}$ .

STEP 16

The consumers' surplus is $\$\textbf{48341.90}$ . The producers' surplus is $\$\textbf{68921.00}$ .

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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**: 4286.55−1.05x2=1.5x24286.55 - 1.05x^2 = 1.5x^24286.55−1.05x2=1.5x2.This is where the price consumers are willing to pay matches the price producers are willing to accept.

STEP 4

Now, let's **solve for** xxx.Add 1.05x21.05x^21.05x2 to both sides of the equation: 4286.55=2.55x24286.55 = 2.55x^24286.55=2.55x2.

STEP 5

Divide both sides by **2.55**: x2=4286.552.55=1681x^2 = \frac{4286.55}{2.55} = 1681x2=2.554286.55​=1681.

STEP 6

Take the **square root** of both sides to find the equilibrium quantity: x=1681=41x = \sqrt{1681} = 41x=1681​=41.So, the equilibrium quantity is x=41x = \textbf{41}x=41 leaf blowers.

STEP 7

STEP 8

The **consumers' surplus** is the area between the **demand curve** and the **equilibrium price**, integrated from 0 to the **equilibrium quantity**.Think of it as the total amount consumers saved compared to what they were willing to pay!

STEP 9

The formula for consumers' surplus is: ∫041(4286.55−1.05x2)dx−41⋅2521.50\int_0^{41} (4286.55 - 1.05x^2) dx - 41 \cdot 2521.50∫041​(4286.55−1.05x2)dx−41⋅2521.50.

STEP 10

STEP 11

STEP 12

The **producers' surplus** is the area between the **equilibrium price** and the **supply curve**, integrated from 0 to the **equilibrium quantity**.It's the extra money producers made compared to the minimum they were willing to accept!

STEP 13

The formula for producers' surplus is: 41⋅2521.50−∫0411.5x2dx41 \cdot 2521.50 - \int_0^{41} 1.5x^2 dx41⋅2521.50−∫041​1.5x2dx.

STEP 14

**Evaluate the integral**: ∫0411.5x2dx=[1.53x3]041=0.5(41)3=0.5(68921)=34460.5\int_0^{41} 1.5x^2 dx = [\frac{1.5}{3}x^3]_0^{41} = 0.5(41)^3 = 0.5(68921) = 34460.5∫041​1.5x2dx=[31.5​x3]041​=0.5(41)3=0.5(68921)=34460.5.

STEP 15

Now, **subtract** from the rectangle's area: 41⋅2521.50−34460.5=103381.5−34460.5=6892141 \cdot 2521.50 - 34460.5 = 103381.5 - 34460.5 = \textbf{68921}41⋅2521.50−34460.5=103381.5−34460.5=68921.The producers' surplus is $68921\$\textbf{68921}$68921.

STEP 16

The consumers' surplus is $48341.90\$\textbf{48341.90}$48341.90. The producers' surplus is $68921.00\$\textbf{68921.00}$68921.00.

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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: $4286.55 - 1.05x^2 = 1.5x^2$ .
This is where the price consumers are willing to pay matches the price producers are willing to accept.

Now, let's solve for $x$ .
Add $1.05x^2$ to both sides of the equation: $4286.55 = 2.55x^2$ .

Divide both sides by 2.55: $x^2 = \frac{4286.55}{2.55} = 1681$ .

Take the square root of both sides to find the equilibrium quantity: $x = \sqrt{1681} = 41$ .
So, the equilibrium quantity is $x = \textbf{41}$ leaf blowers.

The consumers' surplus is the area between the demand curve and the equilibrium price, integrated from 0 to the equilibrium quantity.
Think of it as the total amount consumers saved compared to what they were willing to pay!

The formula for consumers' surplus is: $\int_0^{41} (4286.55 - 1.05x^2) dx - 41 \cdot 2521.50$ .

The producers' surplus is the area between the equilibrium price and the supply curve, integrated from 0 to the equilibrium quantity.
It's the extra money producers made compared to the minimum they were willing to accept!

The formula for producers' surplus is: $41 \cdot 2521.50 - \int_0^{41} 1.5x^2 dx$ .

Evaluate the integral: $\int_0^{41} 1.5x^2 dx = [\frac{1.5}{3}x^3]_0^{41} = 0.5(41)^3 = 0.5(68921) = 34460.5$ .

Now, subtract from the rectangle's area: $41 \cdot 2521.50 - 34460.5 = 103381.5 - 34460.5 = \textbf{68921}$ .
The producers' surplus is $\$\textbf{68921}$ .

The consumers' surplus is $\$\textbf{48341.90}$ . The producers' surplus is $\$\textbf{68921.00}$ .