A rectangular garden is 20 ft longer than it is wide. Its area is 8000 ft{}^2. What are its dimensions?
Let "w" equal the width
w+20=8000
w^2+20=8000
not sure how to continue
If w = width, then length = w + 20
w(w+20) = 8000
Solve for w, then length.
you have the right idea, but you should have said
w(w+20) = 8000
w^2+20w = 8000
w^2+20w-8000 = 0
(w-80)(w+100) = 0
...
How do I isolate w ?
so w=80 and w=-100 ?
If two numbers multiply to produce a product of zero, then one of them must be zero.
So, if (w-80)(w+100)=0, then either
w-80 = 0
w=80
or
w+100 = 0
w = -100
Now pick the value that makes sense for this problem. Then check your conclusion by plugging in the value and seeing whether it produces a rectangle of the correct area.
To solve this problem, we need to set up an equation based on the given information and then solve for the unknown variables.
Let's use "w" to represent the width of the rectangular garden. According to the problem, the length of the garden is 20 ft longer than the width, which means the length can be expressed as "w + 20".
Next, we know that the area of a rectangle is given by the formula A = length × width. In this case, the area is given as 8000 ft^2. So, we can set up the equation:
w × (w + 20) = 8000
To solve this equation, we need to expand the brackets:
w^2 + 20w = 8000
Now, we have a quadratic equation. Let's rearrange it to get it in standard form:
w^2 + 20w - 8000 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring might not be easy in this case since the numbers involved are large. So, let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = 20, and c = -8000. Plugging these values into the quadratic formula, we get:
w = (-20 ± √(20^2 - 4(1)(-8000))) / (2(1))
Simplifying further:
w = (-20 ± √(400 + 32000)) / 2
w = (-20 ± √(32400)) / 2
w = (-20 ± 180) / 2
Now, we have two possible solutions for w. Let's calculate them separately:
For w = (-20 + 180) / 2 = 160/2 = 80
For w = (-20 - 180) / 2 = -200/2 = -100
Since a negative width doesn't make sense, we discard the negative value. Therefore, the width of the rectangular garden is 80 ft.
To find the length, we can substitute this value back into the equation we formed:
length = w + 20 = 80 + 20 = 100 ft
So, the dimensions of the rectangular garden are 80 ft (width) and 100 ft (length).