MATH SOLVE

4 months ago

Q:
# Miriam has 62 feet of fencing to make a rectangular vegetable garden. Which dimensions will give Miriam the garden with greatest area? The diagrams are not to scale.

Accepted Solution

A:

Perimeter: P=62 feet

P=2(b+h)

62=2(b+h)

Dividing both sides of the equation by 2:

62/2=2(b+h)/2

31=b+h

b+h=31 (1)

Area: A=bh (2)

Isolating h in equation (1)

(1) b+h=31→b+h-b=31-b→h=31-b (3)

Replacing h by 31-b in equation (2)

(2) A=bh

A=b(31-b)

A=31b-b^2

To maximize the area:

A'=0

A'=(A)'=(31b-b^2)'=(31b)'-(b^2)'=31-2b^(2-1)→A'=31-2b

A'=0→31-2b=0

Solving for b:

31-2b+2b=0+2b

31=2b

Dividing both sides by 2:

31/2=2b/2

31/2=b

b=31/2=15.5

Replacing b by 31/2 in equation (3)

h=31-b

h=31-31/2

h=(2*31-31)/2

h=(62-31)/2

h=31/2

The dimensions are 31/2 ft x 31/2 ft = 15.5 ft x 15.5 ft

The area with these dimensions is: A=(15.5 ft)(15.5 ft)→A=240.25 ft^2

These dimensions are not in the options

1) The first option has an area of: A=(18 ft)(13 ft)→A=234 ft^2

2) The second option has an area of: A=(19 ft)(12 ft)→A=228 ft^2

3) The third option has an area of: A=(17 ft)(14 ft)→A=238 ft^2

The third option has the largest area.

Answer: Third option

P=2(b+h)

62=2(b+h)

Dividing both sides of the equation by 2:

62/2=2(b+h)/2

31=b+h

b+h=31 (1)

Area: A=bh (2)

Isolating h in equation (1)

(1) b+h=31→b+h-b=31-b→h=31-b (3)

Replacing h by 31-b in equation (2)

(2) A=bh

A=b(31-b)

A=31b-b^2

To maximize the area:

A'=0

A'=(A)'=(31b-b^2)'=(31b)'-(b^2)'=31-2b^(2-1)→A'=31-2b

A'=0→31-2b=0

Solving for b:

31-2b+2b=0+2b

31=2b

Dividing both sides by 2:

31/2=2b/2

31/2=b

b=31/2=15.5

Replacing b by 31/2 in equation (3)

h=31-b

h=31-31/2

h=(2*31-31)/2

h=(62-31)/2

h=31/2

The dimensions are 31/2 ft x 31/2 ft = 15.5 ft x 15.5 ft

The area with these dimensions is: A=(15.5 ft)(15.5 ft)→A=240.25 ft^2

These dimensions are not in the options

1) The first option has an area of: A=(18 ft)(13 ft)→A=234 ft^2

2) The second option has an area of: A=(19 ft)(12 ft)→A=228 ft^2

3) The third option has an area of: A=(17 ft)(14 ft)→A=238 ft^2

The third option has the largest area.

Answer: Third option