Exercise 26 Page 935

Practice makes perfect

a We want to estimate the approximate combined area of the spinach and corn fields.

Let's take a closer look at the corn and spinach fields. To simplify our solution, we can say that each plot has a side length of $1 .$

We can see that the corn field is a rectangle with length

ℓ = 7

and width

w = 4 .

The spinach field is a trapezoid with bases of length

b_{1} = 6

and

b_{2} = 7,

and height of

h = 6 .

Let's recall the formulas for areas

A_{R}

and

A_{T}

of these figures.

Area of a Rectangle : Area of a Trapezoid : A_{R} = ℓ w A_{T} = \frac{1}{2} h (b_{1} + b_{2})

Let's use our values of

ℓ = 7

and

w = 4

to find the area of the corn field.

A_{R} = ℓ w

SubstituteII

$ℓ = 7$ , $w = 4$

A_{R} = (7) (4)

Multiply

A_{R} = 28

Now we will substitute

6

for

h,

6

for

b_{1},

and

7

for

b_{2}

into the formula for the area of a trapezoid to find the area of the spinach field.

A_{T} = \frac{1}{2} h (b_{1} + b_{2})

SubstituteValues

Substitute values

A_{T} = \frac{1}{2} (6) (6 + 7)

AddTerms

Add terms

A_{T} = \frac{1}{2} (6) (13)

Multiply

A_{T} = 39

The corn field consists of

28

plots and the spinach area consists of

39

plots. We can now estimate the combined area of both fields.

Combined Area = 28 + 39 ⇕ Combined Area = 67

The spinach and corn field have a combined area of approximately

67

plots.

b We want to find the probability that a randomly selected plot is used to plant soybeans. To do so we will use geometric probability. The probability of a randomly selected plot being used to plant soybeans is equal to the ratio of the area used to grow soybeans to the area of the entire farm.

P (the plot is being used to grow soybeans) = \frac{Area of the Soybean Field}{Area of the Farm}

Let's find the area of the field used to grow soybeans and the area of the entire farm. Just like in Part A, we will say that each plot has a side length of

1 .

As we can see from the diagram, the farm is a rectangle with length

ℓ = 17

and width

w = 10 .

Let's recall the formula for the area

A

of a rectangle with length

ℓ

and width

w .

A = ℓ w

We can now calculate the area of the farm.

A = ℓ w

SubstituteII

$ℓ = 17$ , $w = 10$

A = (17) (10)

Multiply

A = 170

The entire farm contains

170

plots. Let's now take a closer look at the soybean field. We can think of the field as a rectangle and a trapezoid.

The total area of the soybean field is the sum of the area of a rectangle with length

ℓ = 5

and width

w = 4,

and the area of a trapezoid with height

h = 2

and bases of lengths

b_{1} = 2.5

and

b_{2} = 5 .

Let's recall the formulas for the areas of these figures.

Area of a Rectangle : Area of a Trapezoid : A_{R} = ℓ w A_{T} = \frac{1}{2} h (b_{1} + b_{2})

Let's calculate the areas of both parts of the field, starting with the rectangular part.

A_{R} = ℓ w

SubstituteII

$ℓ = 5$ , $w = 4$

A_{R} = (5) (4)

Multiply

A_{R} = 20

The rectangular part of the soybean field contains

20

plots. Let's now calculate the area of the trapezoidal part.

A_{T} = \frac{1}{2} h (b_{1} + b_{2})

SubstituteValues

Substitute values

A_{T} = \frac{1}{2} (2) (2.5 + 5)

AddTerms

Add terms

A_{T} = \frac{1}{2} (2) (7.5)

Multiply

A_{T} = 7.5

The trapezoidal part of the soybean field contains

7.5

plots. Now we can calculate the area of the soybean field by adding the area of the rectangular part and the area of the trapezoidal part.

Area of the Soybean Field = 20 + 7.5 ⇕ Area of the Soybean Field = 27.5

Finally, to find the probability of a randomly chosen plot being used to grow soybeans we will calculate the ratio of the area of the soybean field to the area of the farm.