Q: A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. $$800\ \text{ft} of $$fencing is used. Find the dimensions of the playground that maximize the total enclosed area.

$$140\ \text{ft} \times 210\ \text{ft}$$

Q: A rectangular garden is to be constructed using 100 meters of fencing. If one side of the garden is against a wall and does not require fencing, what are the dimensions that maximize the area?

$$25\ \text{m} \times 50\ \text{m}$$

Q: A company wants to design a box with a square base and open top that will hold $$32\ \text{ft}^3. $$What dimensions will minimize the amount of material used?

$$4\ \text{ft} \times 4\ \text{ft} \times 2\ \text{ft}$$

Q: A farmer wants to fence a rectangular field adjacent to a river, using no fence along the river. If the farmer has 240 meters of fencing, what dimensions will maximize the area?

$$60\ \text{m} \times 120\ \text{m}$$

Q: A manufacturer wants to make a cylindrical can that will hold $$500\ \text{cm}^3 of $$liquid. What dimensions will minimize the amount of material used?

$$r = 5.42\ \text{cm},\ h = 5.42\ \text{cm}$$

Q: A rectangular pen is to be built using 100 meters of fencing. If the pen is to be divided into two equal sections by a fence parallel to one of the sides, what are the dimensions that maximize the area?

$$20\ \text{m} \times 30\ \text{m}$$

Q: A company wants to design a box with a square base and closed top that will hold $$64\ \text{ft}^3. $$What dimensions will minimize the amount of material used?

$$4\ \text{ft} \times 4\ \text{ft} \times 4\ \text{ft}$$

Question 1

A developer wants to enclose a rectangular grass lot that borders a city street for parking. If the developer has 280 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?

Accepted Answer

$$19,600\ 	ext{ft}^2$$

Question 2

A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow.

Accepted Answer

$$4.5\ 	ext{inches}$$

Question 3

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. $$800\ 	ext{ft} of $$fencing is used. Find the dimensions of the playground that maximize the total enclosed area.

Accepted Answer

$$140\ 	ext{ft} 	imes 210\ 	ext{ft}$$

Question 4

In one U.S. city, the quadratic function $$f(x) = 0.003x^2$ - 0.4x + 36.4$$ models the median, or average, age, $$x$$, at which men first married years after 1900. In which year was this average age at a minimum?

Accepted Answer

$$1946$$

Question 5

Using the same function $$f(x) = 0.003x^2$ - 0.4x + 36.4$$, what was the average age at first marriage for men in that year? (Round to the nearest tenth.)

Accepted Answer

$$24.2\ 	ext{years}$$

Question 6

A person standing close to the edge on top of a 300-foot building throws a baseball vertically upward. The quadratic function $$h(t) = -16t^2$ + 80t + 300$$ models the ball's height above the ground, $$h$$, in feet, seconds after it was thrown. How many seconds does it take until the ball reaches its maximum height?

Accepted Answer

$$5\ 	ext{seconds}$$

Question 7

Using the same function $$h(t) = -16t^2$ + 80t + 300$$, what is the maximum height reached by the ball?

Accepted Answer

$$500\ 	ext{ft}$$

Question 8

An object is projected vertically upward from the top of a 300-foot building. The quadratic function $$h(t) = -16t^2$ + 80t + 300$$ models the object's height above the ground, $$h$$, in feet, seconds after it was thrown. How many seconds does it take until the object finally hits the ground? Round to the nearest tenth if necessary.

Accepted Answer

$$13.4\ 	ext{seconds}$$

Question 9

A rectangular garden is to be constructed using 100 meters of fencing. If one side of the garden is against a wall and does not require fencing, what are the dimensions that maximize the area?

Accepted Answer

$$25\ 	ext{m} 	imes 50\ 	ext{m}$$

Question 10

A company wants to design a box with a square base and open top that will hold $$32\ 	ext{ft}^3. $$What dimensions will minimize the amount of material used?

Accepted Answer

$$4\ 	ext{ft} 	imes 4\ 	ext{ft} 	imes 2\ 	ext{ft}$$

Question 11

A farmer wants to fence a rectangular field adjacent to a river, using no fence along the river. If the farmer has 240 meters of fencing, what dimensions will maximize the area?

Accepted Answer

$$60\ 	ext{m} 	imes 120\ 	ext{m}$$

Question 12

A manufacturer wants to make a cylindrical can that will hold $$500\ 	ext{cm}^3 of $$liquid. What dimensions will minimize the amount of material used?

Accepted Answer

$$r = 5.42\ 	ext{cm},\ h = 5.42\ 	ext{cm}$$

Question 13

A rectangular field is to be fenced off along a straight road using 400 meters of fencing. If the side along the road is not to be fenced, what is the maximum area that can be enclosed?

Accepted Answer

$$40,000\ 	ext{m}^2$$

Question 14

A ball is thrown upward from the ground with an initial velocity of $$64\ 	ext{ft/s}. $$Its height after $$t$$ seconds is given by $$h(t) = -16t^2$ + 64t. $$What is the maximum height reached by the ball?

Accepted Answer

$$128\ 	ext{ft}$$

Question 15

A rectangular pen is to be built using 100 meters of fencing. If the pen is to be divided into two equal sections by a fence parallel to one of the sides, what are the dimensions that maximize the area?

Accepted Answer

$$20\ 	ext{m} 	imes 30\ 	ext{m}$$

Question 16

A company wants to design a box with a square base and closed top that will hold $$64\ 	ext{ft}^3. $$What dimensions will minimize the amount of material used?

Accepted Answer

$$4\ 	ext{ft} 	imes 4\ 	ext{ft} 	imes 4\ 	ext{ft}$$

Question 17

A ball is thrown upward from the top of a $$100\ 	ext{ft}$$ building with an initial velocity of $$48\ 	ext{ft/s}. $$Its height after $$t$$ seconds is given by $$h(t) = -16t^2$ + 48t + 100. $$How long does it take for the ball to hit the ground?

Accepted Answer

$$6.0\ 	ext{seconds}$$

Precalculus Applications: Quadratic Functions, Optimization, and Projectile Motion

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