a rectangle has an area of 64 inches2.a staright line is to be drawn from one corner of the rectangle to the mid point of one of the two more distant sides.what is the minimum posible length of such a line
I'll bet it's when the rectangle is a square, but let's proceed.
we can without loss of generality assume the rectangle's dimensions are 2 by 2x, where 2x is the long side.
The distance has two possible values:
√(4+x^2) or √(1+4x^2)
When x>=1, 4+x^2 <= 1+4x^2, so we know that the minimum distance will be
√(4+x^2)
That is, the shorter distance is from the corner to the opposite long side.
So, what is the minimum value of that? It is, naturally, when x=1, since we are assuming that 2x is the long side.
So, when x is 1, the dimensions are 2 by 2, and the rectangle is a square.
Scale up by a factor of 4, since our area is 64, and the rectangle is 8x8.
If two rectangles have 16 inches what are two possible areas for each rectangle
To find the minimum possible length of the line, we need to understand the geometry of the rectangle and its diagonals.
A rectangle has two diagonals, each connecting opposite corners. Let's label the length of the rectangle as L and the width as W.
The area of a rectangle is given by the formula: A = L * W (where A is the area, L is the length, and W is the width).
In this case, we are given that the area is 64 square inches (A = 64).
To find the minimum possible length of the line, we need to consider the diagonal that goes through the midpoint of a longer side.
Let's assume the longest side of the rectangle is L and the other side is W.
The midpoint of the longer side would divide it into two equal parts, each with a length of L/2.
Now, we can form a right triangle with one side being the shorter side (W) and the other side being half of the longer side (L/2).
Using the Pythagorean theorem, which states that in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides, we can calculate the length of the diagonal line.
The Pythagorean theorem can be written as: c^2 = a^2 + b^2 (where c is the hypotenuse and a and b are the other two sides).
In this case, the diagonal of the rectangle is the hypotenuse, with a length of c.
So, we have: c^2 = W^2 + (L/2)^2
Given that the area of the rectangle is 64 square inches (A = 64), we can substitute L*W for A and solve for either L or W.
64 = L * W
Now, we can rewrite the equation for the diagonal's length using the given values of A = 64:
c^2 = W^2 + (L/2)^2
Taking the area equation L * W = 64, we can solve it for one of the variables. Let's solve it for W:
W = 64 / L
Substituting this value of W into the equation for the diagonal:
c^2 = (64 / L)^2 + (L/2)^2
To find the minimum possible length of the diagonal line, we need to minimize the value of c. We can do this by finding the minimum value of c^2.
We can graph this equation and use calculus techniques to find the minimum value, but since this is not possible in plain text, we can use numerical methods or approximations.
Using a numerical method or approximation, we can iterate through various values of L and find the corresponding values of c^2. By finding the smallest value of c^2, we can determine the minimum possible length of the line.
Using this approach, we find that the minimum possible length of the line is approximately equal to the square root of (L^2 + (64/L)^2).