A garden measuring 12m by 16m is to gave a pedestrian pathway installed all around it. The pathway is the same width all the way around the garden. The pathway and the garden have a total area of 258m^2. What will be the width of the pathway? ( I know the answer is 1.5 I just don't know how to get that. Please explain as you go)
width of pathway = x m
width of whole thing = 12+2x m
length of whole thing = 16+2x
(12+2x)(16+2x) = 258
192 + 56x + 4x^2 = 258
2x^2 + 28x - 33 = 0
x = (-28 +- sqrt(1048))/4
= 1.093 or a negative
the path should be appr 1.09 m wide all around
check:
(12+ 2.18)(16+2.18)
= 14.18 x 18.18
= 257.79
close enough to our accuracy
The correct answer to the way you typed it is 1.093
Looking at the first Related Question below, I noticed that their area was 285 m^2
Did you transpose digits?
If so, just adjust my solution.
Okay thank you so much!!!
To find the width of the pathway, we need to subtract the area of the garden from the total area of the garden and the pathway combined.
Given that the total area is 258m^2 and the garden is 12m by 16m, we can find the area of the garden by multiplying the length and width:
Area of the garden = Length * Width
= 12m * 16m
= 192m^2
To find the area of the pathway, we subtract the area of the garden from the total area:
Area of the pathway = Total area - Area of the garden
= 258m^2 - 192m^2
= 66m^2
Now, since the pathway surrounds the garden on all sides, the width of the pathway will be the same on all sides. Let's assume the width of the pathway is "x".
We can now calculate the total width of the garden and the pathway together:
Total width = Garden width + 2 * Width of the pathway
= 16m + 2 * x
Similarly, we can calculate the total length of the garden and pathway together:
Total length = Garden length + 2 * Width of the pathway
= 12m + 2 * x
Now, to find the area of the garden and pathway combined, we can multiply the total width and total length:
Area of the garden and pathway combined = Total width * Total length
= (16m + 2 * x) * (12m + 2 * x)
Since we know that the area of the garden and pathway combined is 258m^2, we can equate the equation above to 258:
(16m + 2 * x) * (12m + 2 * x) = 258
Now, we can solve this equation to find the value of x, which is the width of the pathway.
Simplifying the equation:
192m^2 + 32m * x + 4 * x^2 = 258
Rearranging terms:
4 * x^2 + 32m * x + 192m^2 - 258 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± sqrt(b^2 - 4 * a * c)) / (2 * a)
In this case, the coefficient of x^2 is 4, the coefficient of x is 32m, and the constant term is 192m^2 - 258.
Plugging in these values into the quadratic formula, we get:
x = (-32m ± sqrt((32m)^2 - 4 * 4 * (192m^2 - 258))) / (2 * 4)
Simplifying further, we get:
x = (-32m ± sqrt(1024m^2 - 3072m^2 + 1032)) / 8
= (-32m ± sqrt(-2048m^2 + 1032)) / 8
At this point, we can see that the value inside the square root is negative since -2048m^2 + 1032 evaluates to a negative number. This means that the quadratic equation has no real solutions, which tells us that there is no valid pathway width that satisfies the given conditions.
However, you mentioned that the answer is 1.5. It seems that there might be an error in the given information or the calculation process. Please double-check the question or provide any additional information if available.