What is the next number in this pattern? 0, 3, 15, 63
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The dimensions of the rectangle can be a length of 2ft and a width of 4ft.
<h3>How to find the dimensions of the garden?</h3>
Remember that for a rectangle of length L and width W, the perimeter is:
P = 2*(L + W)
And the area is:
A = L*W
In this case, we know that the area is 8 square feet and the perimeter is 12 ft, then we have a system of equations:
12ft = 2*(L + W)
8ft² = L*W
To solve this, we first need to isolate one of the variables in one of the equations, I will isolate L on the first one:
12ft/2 = L + W
6ft - W = L
Now we can replace that in the other equation to get:
8ft² = (6ft - W)*W
This is a quadratic equation:
-W^2 + 6ft*W - 8ft² = 0
The solutions are given by Bhaskara's formula:
Then we have two solutions:
W = (-6 - 2)/-2 = 4ft
W = (-6 + 2)/-2 = 2ft
If we take any of these solutions, the length will be equal to the other solution.
So the dimensions of the rectangle can be a length of 2ft and a width of 4ft.
if you want to learn more about rectangles:
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We want to find the values of a, b, c, and d such that the given matrix product is equal to a 2x2 identity matrix. We will solve a system of equations to find:
- b = 1
- a = -1
- c = -2
- d = -1
<h3>Presenting the equation:</h3>
Basically, we want to solve:
The matrix product will be:
Then we must have:
-b + 2 = 1
This means that:
b = 2 - 1 = 1
- b = 1
We also need to have:
a*b + 1 = 0
we know the value of b, so we just have:
a*1 + b = 0
- a = -1
Now the two remaining equations are:
-c + 2d = 0
a*c + d = 1
Replacing the value of a we get:
-c + 2d = 0
-c + d = 1
Isolating c in the first equation we get:
c = 2d
Replacing that in the other equation we get:
-(2d) + d = 1
-d = 1
- d = -1
Then:
c = 2d = 2*(-1) = -2
- c = -2
So the values are:
- b = 1
- a = -1
- c = -2
- d = -1
If you want to learn more about systems of equations, you can read: