Question

The ratio of the length to width of a golden rectangle is (1+ square root of five):2. The dimensions of a garden )shown below) form a golden rectangle.a. Find the length L of the garden round your answer to the nearest footb. If the gardener wants to plant a tomato for every 3 square feet how many tomato plants would be needed.c. Write and simplify an expression to find the area of the path that surrounds the garden.

Answer 1

a) Length = 29 ft

b) 251 tomato plants will be needed.

c) Area of the path = 4x² + 110x

Step-by-step explanation

The ratio of the length of the garden to the width of the garden is

(1 + √5) : 2

From the image, we are given that the

Width = 26 ft

Length = ?

And a path that is x feet wide surrounds the garden.

a) Find the Length of the garden.

Length = ?

Width = 26 ft.

We can just find the Length by expressing the ratios as fractions

Length : Width = (1 + √5) : 2

L : 26 = (1 + √5) : 2

`\begin{gathered} (L)/(26)=\frac{(1+\sqrt[]{5})}{2} \\ \text{Cross multiply} \\ 2L=26(1+\sqrt[]{5}) \\ Divide\text{ both sides by 2} \\ L=13(1+\sqrt[]{5}) \end{gathered}`

L = 13 (1 + √5)

L = 13 (1 + 2.236)

L = 13 (3.236)

L = 29.07 ft. = 29 ft. to the nearest foot

b) The gardener wants to plant a tomato for every 3 square feet.

We are told to find the number of tomatoes that can be planted on this garden.

So, we need to first find the area of the garden,

Area = Length × Width

Area = 29 × 26

Area = 754 ft²

Let the number of tomatoes that can be planted on this 754 ft² of the garden be x

1 tomato = 3 ft²

x tomatoes = 754 ft²

We can form a mathematical relationship by cross multiplying

(x) (3) = (1) (754)

3x = 754

Divide both sides by 3

(3x/3) = (754/3)

x = 251.3 tomatoes = 251 tomatoes to the nearest whole number.

c) The garden plus the path around it form a rectangle too

Area = Length × Width

Length = (29 + 2x) ft.

Width = (26 + 2x) ft.

Area = Length × Width

Area = (29 + 2x) × (26 + 2x)

Area = (29 + 2x) (26 + 2x)

Area = 29 (26 + 2x) + 2x (26 + 2x)

Area = 754 + 58x + 52x + 4x²

Area = 4x² + 110x + 754

Area of the path = (Area of the garden and path) - (Area of the garden)

Area of the path = (4x² + 110x + 754) - 754

Area of the path = 4x² + 110x + 754 - 754

Area of the path = 4x² + 110x

Hope this Helps!!!