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price = p

demand = x

we need to write demand as a function of price. So f(p) = x.

1.

p = 42 - 0.4x ; 0 ≤ x ≤ 105

for x = 0, p = 42

for x = 105, p = 42 - 0.4×105 = 42 - 42 = 0

p = 42 - 0.4x

⇒ p - 42 = -0.4x

⇒ x = -(p-42)/0.4

⇒ x = 105 - 2.5p

⇒

2.

p = 125 - 0.02x ; 0 ≤ x ≤ 6250

for x = 0, p = 125

for x = 6250, p = 125 - 0.02×6250 = 125 - 125 = 0

p =125 - 0.02x

⇒ p - 125 = -0.02x

⇒ -x = (p - 125)/0.02

⇒ -x = 50p - 6250

⇒ x = 6250 - 50p

⇒

3.

p = 50 - 0.5x² ; 0 ≤ x ≤10

for x = 0, p = 50

for x = 10, p = 50 - 0.5×10² = 50 - 50 = 0

p =50 - 0.5x²

⇒ p - 50 = -0.5x²

⇒ 0.5x² = 50 - p

⇒ x² = (50 - p)/0.5 = 100 - 2p

⇒ x = √(100 - 2p)

⇒

4.

p = 180 - 0.8x² ; 0 ≤ x ≤15

for x = 0, p = 180

for x = 15, p = 180 - 0.8×15² = 180-180 = 0

p =180 - 0.8x²

⇒ p - 180 = -0.8x²

⇒ 0.8x² = 180 - p

⇒ x² = (180 - p)/0.8 = 225 - 1.25p

⇒ x = √( 225 - 1.25p)

⇒

5.

p = 25e^{-x/20} ; 0≤x≤20

for x = 0, p = 25

for x = 20, p = 25e^{-20/20} = 25/e

p = 25e^{-x/20}

⇒ p = 25/e^{x/20}

⇒ e^{x/20} = 25/p

⇒ log( e^{x/20} ) = log (25/p)

⇒ x/20 = log (25/p)

⇒ x = 20 log (25/p)

⇒

6.

p = 45 - e^{x/4} ; 0≤x≤12

for x = 0, p = 45-1 = 44

for x = 12, p = 45 - e^{12/4} = 45- e³

p = 45 - e^{x/4}

⇒ p - 15= e^{x/4}

⇒ e^{x/4} = 45 - p

⇒ log( e^{x/4} ) = log (45 - p)

⇒ x/4 = log (45 - p)

⇒ x = 4 log (45 - p)

⇒

7.

p =80 - 10 ln x; 1 ≤ x ≤ 30

for x = 1, p = 80 - 0 = 80

for x = 30, p = 80 - 10 ln (30) = 10 (8 - ln 30)

p = 80 - 10 ln x

⇒ p - 80 = - 10 ln x

⇒ 10 ln x = 80-p

⇒ ln x = (80-p)/10 = 8 - 0.1p

⇒ x = e^{8 - 0.1p}

⇒

8.

p =ln(500 - 5x); 0 ≤ x ≤ 90

for x = 0, p =ln 500

for x = 90, p = ln(500 - 5×90) = ln 50

p = ln(500 - 5x)

⇒ ln(500 - 5x) = p

⇒ 500 - 5x = e^p

⇒ -5x = e^p - 500

⇒ x = (e^p - 500)/(-5) = 100 - 0.2e^p

⇒

demand = x

we need to write demand as a function of price. So f(p) = x.

1.

p = 42 - 0.4x ; 0 ≤ x ≤ 105

for x = 0, p = 42

for x = 105, p = 42 - 0.4×105 = 42 - 42 = 0

p = 42 - 0.4x

⇒ p - 42 = -0.4x

⇒ x = -(p-42)/0.4

⇒ x = 105 - 2.5p

⇒

**f(p) = 105 - 2.5p ; 0 ≤ p ≤ 105.**2.

p = 125 - 0.02x ; 0 ≤ x ≤ 6250

for x = 0, p = 125

for x = 6250, p = 125 - 0.02×6250 = 125 - 125 = 0

p =125 - 0.02x

⇒ p - 125 = -0.02x

⇒ -x = (p - 125)/0.02

⇒ -x = 50p - 6250

⇒ x = 6250 - 50p

⇒

**f(p) = 6250 - 50p ; 0 ≤ p ≤ 125.**3.

p = 50 - 0.5x² ; 0 ≤ x ≤10

for x = 0, p = 50

for x = 10, p = 50 - 0.5×10² = 50 - 50 = 0

p =50 - 0.5x²

⇒ p - 50 = -0.5x²

⇒ 0.5x² = 50 - p

⇒ x² = (50 - p)/0.5 = 100 - 2p

⇒ x = √(100 - 2p)

⇒

**f(p) = √(100 - 2p); 0 ≤ p ≤ 50.**4.

p = 180 - 0.8x² ; 0 ≤ x ≤15

for x = 0, p = 180

for x = 15, p = 180 - 0.8×15² = 180-180 = 0

p =180 - 0.8x²

⇒ p - 180 = -0.8x²

⇒ 0.8x² = 180 - p

⇒ x² = (180 - p)/0.8 = 225 - 1.25p

⇒ x = √( 225 - 1.25p)

⇒

**f(p) = √(****225 - 1.25p); 0 ≤ p ≤ 180.**5.

p = 25e^{-x/20} ; 0≤x≤20

for x = 0, p = 25

for x = 20, p = 25e^{-20/20} = 25/e

p = 25e^{-x/20}

⇒ p = 25/e^{x/20}

⇒ e^{x/20} = 25/p

⇒ log( e^{x/20} ) = log (25/p)

⇒ x/20 = log (25/p)

⇒ x = 20 log (25/p)

⇒

**f(p) = 20 log (25/p); 0 ≤ p ≤ 25/e**6.

p = 45 - e^{x/4} ; 0≤x≤12

for x = 0, p = 45-1 = 44

for x = 12, p = 45 - e^{12/4} = 45- e³

p = 45 - e^{x/4}

⇒ p - 15= e^{x/4}

⇒ e^{x/4} = 45 - p

⇒ log( e^{x/4} ) = log (45 - p)

⇒ x/4 = log (45 - p)

⇒ x = 4 log (45 - p)

⇒

**f(p) = 20 log (25/p); 45-e**³**≤ p ≤ 44**7.

p =80 - 10 ln x; 1 ≤ x ≤ 30

for x = 1, p = 80 - 0 = 80

for x = 30, p = 80 - 10 ln (30) = 10 (8 - ln 30)

p = 80 - 10 ln x

⇒ p - 80 = - 10 ln x

⇒ 10 ln x = 80-p

⇒ ln x = (80-p)/10 = 8 - 0.1p

⇒ x = e^{8 - 0.1p}

⇒

**f(p) = e^{8 - 0.1p}; 10(8- ln 30) ≤ p ≤ 80**8.

p =ln(500 - 5x); 0 ≤ x ≤ 90

for x = 0, p =ln 500

for x = 90, p = ln(500 - 5×90) = ln 50

p = ln(500 - 5x)

⇒ ln(500 - 5x) = p

⇒ 500 - 5x = e^p

⇒ -5x = e^p - 500

⇒ x = (e^p - 500)/(-5) = 100 - 0.2e^p

⇒

**f(p) = 100 - 0.2e^p; ln 50 ≤ p ≤ ln 500**