Aplikasi Turunan

KD 6.4. APLIKASI TURUNANcari model math susun selesaikan !Jadwal Ulangan KD 6.4 (terakhir)11 IPA 1: Kamis, 19 Mei 201111 IPA 2: Jumat, 20 Mei 201111 IPA 3: Jumat, 20 Mei 2011

Turunan I dapat dipakai untuk optimasi fungsi, untuk menentukan nilai maks atau min fungsi. Contoh 1: hal. 355 bawahJumlah 2 bilangan adalah 8. Tentukan kedua bil. itu agar jumlah kuadrat keduanya menjadi minimum.Jawab: Misal kedua bil. itu x dan y x + y = 8 y = 8 x Jumlah kuadrat: J = x2 + y2 J = x2 + (8 x)2 = 2x2 16x + 64 Agar minimum JI = 0 4x 16 = 0 x = 4 4 + y = 8 y = 4 Jadi, kedua bil. itu adalah 4 dan 4

Contoh 2: hal. 356 Seutas kawat (16 cm) dipotong mjd 2 bagian. Potongan I (8x cm) dibuat segi4 ukuran 3x dan x, potongan II dibuat persegi. Tentukan total luas minimum keduanya. Jawab: Total panjang segi4 = 8x sisa kawat = 16 8x Sisi persegi = (16 8x) / 4 = 4 2x Luas total: L = 3x . x + (4 2x)2 = 7x2 16x + 16LI = 0 14x 16 = 0 x = 8/7

Contoh 3: hal. 357 Sebuah tabung (radius r2, tinggi h2) dimasukkan kedalam kerucut (radius r1, tinggi h1). Tentukan vol. maks tabung itu. Jawab: Pakai perbandingan tangent :

1. Find two non negative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. 2. Build a rectangular pen with three parallel partitions using 500 cm of fencing. What dimensions will maximize the total area of the pen ? 3. An open rectangular box with square base is to be made from 48 dm2 of material. What dimensions will result in a box with the largest possible volume ? 4. A container in the shape of a right circular cylinder with no top has surface area 3 dm2. What height h and base radius r will maximize the volume of the cylinder ? 5. A sheet of cardboard 3 dm by 4 dm. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume ? PROBLEMS from internet

6. Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse. 8. A cylindrical can is to hold 20 m3. The material for the top and bottom costs Rp 10.000/m2 and material for the side costs Rp 8.000/m2. Find the radius r and height h of the most economical can. 9. Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2. 10. What angle between two edges of length 3 will result in an isosceles triangle with the largest area ? segitiga samakaki11. Car B is 30 km directly east of car A and begins moving west at speed 90 km/h. At the same moment car A begins moving north at 60 km/h. What will be the minimum distance between the cars and at what time t does the minimum distance occur ?

12. A rectangular piece of paper is 12 cm high and 6 cm wide. The lower right tbootom-hand corner is folded over so as to reach the leftmost edge of the paper.13. What positive number added to its reciprocal gives the minimum sum? 14. The sum of two numbers is k. Find the minimum value of the sum of their squares. 15. The sum of two numbers is p. Find the minimum value of the sum of their cubes. 16. The sum of two positive numbers is 4. Find the smallest value possible for the sum of the cube of one number and the square of the other.

17. Find two numbers whose sum is a, if the product of one to the square of the other is to be a minimum.18. Find two numbers whose sum is a, if the product of the square of one by the cube of the other is to be a maximum.

19. A rectangular field of given area is to be fenced off along the bank of a river. If no fence is needed along the river, what is the shape of the rectangle requiring the least amount of fencing?

23. The perimeter of an isosceles triangle is p cm. Find its maximum area.