EXPLANATION
Let's see the facts:
[tex]\text{Ratio}=\frac{1+\sqrt[]{5}}{2}[/tex]Dimenstions of the garden:
[tex]\text{Width = 26 f}eet[/tex][tex]\text{Path = x}[/tex]The length of the garden is given by the following relationship:
[tex]\text{length}=\text{ratio}\cdot\text{width}[/tex]Replacing terms:
[tex]\text{length}=\frac{1+\sqrt[]{5}}{2}\cdot26=13(1+\sqrt[]{5})[/tex]Applying the distributive property:
[tex]\text{length =13 + 13 }\sqrt[]{5}[/tex]Simplifying:
[tex]\text{length = 42.06 }\approx42\text{ fe}et[/tex]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:
[tex]\text{Number of plants of tomato=}\frac{1092ft^2}{3ft^2}=364\text{ plants}[/tex]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:
[tex]\text{Area}_{\text{path}}=\text{Area of the outer surface- Area of the garden}[/tex]The area of the outer surface can be obtained by the following relationship:
[tex]\text{Area}_{\text{outer}}=\text{length}_{\text{outer}}\cdot\text{width}_{\text{outer}}[/tex]Replacing terms:
[tex]\text{Area}_{\text{outer}}=(x+42+x)\cdot(x+26+x)[/tex]Adding like terms:
[tex]\text{Area}_{\text{outer}}=(2x+42)\cdot(2x+26)[/tex]Applying the distributive property:
[tex]\text{Area}_{\text{outer}}=4x^2+52x+84x+68[/tex]Adding like terms:
[tex]\text{Area}_{\text{outer}}=4x^2+126x+68[/tex]Then, subtracting the outer and the garden area give us the area of the path:
[tex]\text{Area}_{\text{path}}=4x^2+126x+68-1092[/tex]Subtracting numbers:
[tex]\text{Area}_{\text{path}}=4x^2+126x-1024[/tex]The expression that represents the area of the path is 4x^2+126x -1024
