A kite frame is to be made from 6 pieces of wood. The four border pieces have been cut 5 and 12. The long center piece is 13. what should the length of the cross pieces be in order to maximize the area of the kite.
If the two top pieces are 5 units each, the bottom pieces are 12 units each, and if the center piece is 13 inches each, you are looking at two right-angled triangles, since
5^2 + 12^2 = 13^2
Thus the length of the cross pieces is fixed as well as the area. So this is not a problem dealing with "maximum" area
the area of each of the two right-angled triangles is
(1/2)(5)12) = 30
so the area of the whole kite is 60 square units
let each height of the triangles be h
(1/2)(13)h = 30
h = 60/13
so the cross piece is 120/13 units long
Well, it seems like we have a "wood" situation here! To maximize the area of the kite, we need to find the length of the cross pieces. Now, let's try not to get "board" with the calculations.
Since the four border pieces have been cut to 5 and 12, we can assume that the kite is a rhombus. The diagonals of a rhombus bisect each other at right angles, forming four congruent right triangles.
The long center piece, which is 13, forms the hypotenuse of two of those right triangles. To maximize the area of the kite, we want these triangles to be as large as possible. In order for that to happen, the two legs of each triangle should be equal.
So, let's call the length of the cross pieces x. This will make the two legs of the right triangles x each. Using the Pythagorean theorem, we can find the length of the longer center piece, which is the hypotenuse:
x^2 + x^2 = 13^2
2x^2 = 169
x^2 = 84.5
x ≈ 9.20 (rounded to two decimal places)
Therefore, the length of the cross pieces should be approximately 9.20 in order to maximize the area of the kite. Now you can "cross" this question off your list and go fly that kite!
To maximize the area of the kite, we need to find the length of the cross pieces.
Let's assume that the lengths of the cross pieces are x and y.
The total length of the border pieces is given as 5 + 12 + 5 + 12 = 34 units.
Since the long center piece is 13 units, the sum of the lengths of the cross pieces needs to be equal to or less than 34 - 13 = 21 units in order to make a closed shape.
Now, let's look at the formula for the area of a kite. The area (A) can be calculated using the formula:
A = 1/2 * d1 * d2
where d1 and d2 are the diagonals of the kite.
In a kite, the diagonals are perpendicular and intersect at a right angle.
Since the long center piece is already fixed at 13 units, let's consider the other diagonal (d2) as 13 units.
Now we have the equation:
A = 1/2 * 13 * d1
To maximize the area of the kite, we need to find the maximum value of d1.
Using Pythagoras' theorem, we know that the sum of the squares of the two cross pieces (x and y) should be equal to the square of d1:
x^2 + y^2 = d1^2
Substituting this expression for d1^2 in the area equation, we get:
A = 1/2 * 13 * (x^2 + y^2)
Now, with the constraint that x + y ≤ 21, we can find the maximum area by finding the maximum value of A using calculus.
Taking the derivative of A with respect to x and y, and setting them to zero, we can find the values of x and y that maximize the area.
After finding the values of x and y, we can substitute them back into the area equation to find the maximum area of the kite.
To determine the length of the cross pieces in order to maximize the area of the kite, we need to understand the properties of a kite. A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. In other words, the four sides of a kite can be divided into two pairs: one pair has equal length, and the other pair also has equal length.
In this case, the four border pieces have lengths of 5 and 12. Let's assume that the side lengths of the kite are labeled as follows: AB = BC = 5, CD = DA = 12. The long center piece is labeled as BE = 13.
To maximize the area of the kite, we need to find the length of the cross pieces that will form the diagonals of the kite. Since diagonals of a kite are perpendicular bisectors of each other, they divide the kite into four right-angled triangles.
To calculate the length of the cross pieces, we can use the Pythagorean theorem for one of the right-angled triangles formed by the diagonals. Let's focus on triangle ABE, where AE and BE are the diagonals of the kite:
AE^2 = AB^2 + BE^2
AE^2 = 5^2 + 13^2
AE^2 = 25 + 169
AE^2 = 194
Taking the square root of both sides, we find:
AE ≈ √194
Therefore, the length of the cross pieces should be approximately equal to √194 in order to maximize the area of the kite.