Video Transcript
A garden is in the shape of a
rectangle. It comprises four rectangular
flower beds separated by paths, as shown in the diagram. The total area of the garden is 468
square meters. Find 𝑥, the width of the
paths.
We note that each rectangular
flower bed has dimensions 10 meters by six meters and that the width of the paths
surrounding the flower beds is unknown. We are given a total area of 468
square meters. And we know that the area of a
rectangle is calculated by multiplying its length times its width. We need to write an equation for
the area of the garden that we could solve for 𝑥. We begin by writing an expression
for the length of the garden.
As shown in the diagram, the length
of the rectangular garden is the sum of 𝑥 plus 10 meters plus 𝑥 plus 10 meters
plus 𝑥. This expression simplifies to three
𝑥 plus 20. The width of the garden is the sum
of 𝑥 plus six meters plus 𝑥 plus six meters plus 𝑥, which simplifies to three 𝑥
plus 12.
Now, we are ready to write the
equation for the area of the garden. We substitute 468 for the area,
three 𝑥 plus 20 for the length, and three 𝑥 plus 12 for the width. If we then use the distributive
property to expand the brackets, we get nine 𝑥 squared plus 36𝑥 plus 60𝑥 plus
240.
Here, it is necessary to simplify
this quadratic equation into standard form so that it can be solved. To write the quadratic in standard
form, we combine the linear terms on the right side of the equation and subtract 468
from each side. The result is the equation zero
equals nine 𝑥 squared plus 96𝑥 minus 228.
We may choose to use any of the
standard methods to solve quadratics, which include factoring, completing the
square, factoring by grouping, or the quadratic formula. Dividing both sides of the equation
by the greatest common factor of three will make this equation a little easier to
solve. Then, we will attempt to factor the
right side of the equation using factoring by grouping. We now have a simplified quadratic
of three 𝑥 squared plus 32𝑥 minus 76.
In order to use the factoring by
grouping method, we will need to find the product of the leading coefficient and the
constant, which is negative 228. Next, we need to find two numbers
that multiply to give negative 228 and add to give the coefficient of the middle
term, which is 32. After considering the many factor
pairs of negative 228, we finally come across a pair of numbers that also add to
32. That is negative six and positive
38. We use these numbers to rewrite the
equation as zero equals three 𝑥 squared minus six 𝑥 plus 38𝑥 minus 76.
We will now take out the greatest
common factor of three 𝑥 from the first two terms and the greatest common factor of
38 in the last two terms. We then get three 𝑥 multiplied by
𝑥 minus two plus 38 multiplied by 𝑥 minus two. Finally, we take out the shared
factor of 𝑥 minus two, giving us 𝑥 minus two times three 𝑥 plus 38.
We recall that for the product of
two factors to be zero, one of the factors must be zero. Hence, either 𝑥 minus two equals
zero or three 𝑥 plus 38 equals zero. By solving each equation, we get 𝑥
equals two and 𝑥 equals negative thirty-eight thirds. Since 𝑥 is the width of a path,
only the positive solution makes sense in the context of the problem. The dimensions of the flower beds
were given in meters, so we also use meters for the value of 𝑥.
In conclusion, the width of the
paths surrounding the four flower beds is two meters.
