Question

Marie wants to buy fencing and sod for her new triangular flower bed. One of the legs of the garden is 7 ft longer than the other and the hypotenuse is 1 ft longer. If sod cost $12.72 per sqft and fencing cost $4.23 per ft. How much would it cost Marie to purchase enough sod and fencing?

Answer 1

let's suppose x is the shortest leg of the triangle

The perimeter of the flower bed is 3x+8 ft

the surface to be coverd by sod would be x×(x+7)/2 sqft

(x+8)(x+8)=(x+7)^2+x^2

x'2+16x+64=x'2+14x+49+x^2

x^2-2x=15

x×(x-2)=15

x=5

(15+8)×$4.23=$97.29 for fencing

32.5sqft×$12.72=$413.4 for the sod

Answer 2

The fencing and sod form a right-angled triangle, with unknown sides. Pythagoras theorem will be used to determine the missing side length.

The total cost to purchase enough sod and fence is $508.50

Let

$Base =x$ the shortest side.

So:

$Other\ leg =7 + x$

$Hypotenuse = 8 + x$

We start by solving for x using Pythagoras theorem

$(8 + x)^2 = (7 + x)^2 + x^2$

Expand

$64 + 8x + 8x + x^2 = 49 + 7x + 7x + x^2 + x^2$

$64 + 16x + x^2 = 49 + 14x + 2x^2$

Rewrite as:

$2x^2 - x^2 + 14x - 16x + 49 - 64 = 0$

$x^2 - 2x -15 = 0$

Expand

$x^2 + 3x - 5x -15 = 0$

Factorize

$x(x + 3) - 5(x +3) = 0$

Factor out x+ 3

$(x - 5) (x +3) = 0$

Split

$x - 5 =0\ or\ x + 3 = 0$

$x = 5\ or\ x =-3$

x can't be negative. So:

$x = 5$

So, we have:

$Base = 5$

$Other\ leg =7+x=7+5=12$

$Hypotenuse =8+x=8+5=13$

The length of fencing needed is the perimeter of the flower bed.

$Fencing = 5 + 12 + 13$

$Fencing = 30$

The fence costs $4.23 per feet; So, the total cost of fencing is:

$Cost = 30 \times \ 4.23$

$Cost = \ 126.9$

The area of Sod is the area of the flower bed.

$Sod = 0.5 \times Base\times Height$

$Sod = 0.5 \times 5 \times 12$

$Sod = 30$

The sod costs $12.72 per square feet; So, the total cost of sod is:

$Cost = 30 \times \ 12.72$

$Cost = \ 381.6$

The total amount is:

$Total = \ 126.9 + \ 381.6$

$Total = \ 508.5$

Hence, $508.5 will be enough for the sod and fencing.

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