83,973 views
23 votes
23 votes
The ratio of the length to the width of a golden rectangle is (1+square root of 5):2.The dimensions of a garden is 26 feet. A path that is x feet wide surrounds the garden.a. Find the length L of the garden round your answer to the nearest foot.b. 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 to expression to find the area of the path that surrounds the garden

The ratio of the length to the width of a golden rectangle is (1+square root of 5):2.The-example-1
User Juanagui
by
2.6k points

1 Answer

9 votes
9 votes

EXPLANATION

Let's see the facts:


\text{Ratio}=\frac{1+\sqrt[]{5}}{2}

Dimenstions of the garden:


\text{Width = 26 f}eet
\text{Path = x}

The length of the garden is given by the following relationship:


\text{length}=\text{ratio}\cdot\text{width}

Replacing terms:


\text{length}=\frac{1+\sqrt[]{5}}{2}\cdot26=13(1+\sqrt[]{5})

Applying the distributive property:


\text{length =13 + 13 }\sqrt[]{5}

Simplifying:


\text{length = 42.06 }\approx42\text{ fe}et

2) If the gardener wants to plant a tomato for every 3 square feet, we need to divide the area of the garden by the required area.

Area of the garden = Length * Width = 26ft * 42 ft = 1092 ft^2

Dividing by the area of each tomato give us the appropiate relationship:


\text{Number of plants of tomato=}(1092ft^2)/(3ft^2)=364\text{ plants}

There would be needed 364 plants of tomato.

3) The area of the path that surrounds the garden could be obtained by the following relationship:


\text{Area}_{\text{path}}=\text{Area of the outer surface- Area of the garden}

The area of the outer surface can be obtained by the following relationship:


\text{Area}_{\text{outer}}=\text{length}_{\text{outer}}\cdot\text{width}_{\text{outer}}

Replacing terms:


\text{Area}_{\text{outer}}=(x+42+x)\cdot(x+26+x)

Adding like terms:


\text{Area}_{\text{outer}}=(2x+42)\cdot(2x+26)

Applying the distributive property:


\text{Area}_{\text{outer}}=4x^2+52x+84x+68

Adding like terms:


\text{Area}_{\text{outer}}=4x^2+126x+68

Then, subtracting the outer and the garden area give us the area of the path:


\text{Area}_{\text{path}}=4x^2+126x+68-1092

Subtracting numbers:


\text{Area}_{\text{path}}=4x^2+126x-1024

The expression that represents the area of the path is 4x^2+126x -1024

User Temor
by
2.8k points