KD 6.4. APLIKASI TURUNAN
cari model math  susun  selesaikan !
Jadwal Ulangan KD 6.4 (terakhir)
11 IPA 1
: Kamis, 19 Mei 2011
11 IPA 2
: Jumat, 20 Mei 2011
11 IPA 3
: Jumat, 20 Mei 2011
Turunan I dapat dipakai untuk optimasi fungsi,
 untuk menentukan nilai maks atau min fungsi.
Contoh 1: hal. 355 bawah
Jumlah 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
4+y=8
Jadi, kedua bil. itu adalah 4 dan 4 


x=4

y=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:
x
Sisi persegi = (16 – 8x) / 4
= 4 – 2x
4 – 2x
4 – 2x
3x
Total panjang segi4 = 8x  sisa kawat = 16 – 8x
Luas total: L = 3x . x + (4 – 2x)2 = 7x2 – 16x + 16
LI = 0

14x – 16 = 0
2

x = 8/7
6
8
8
2
Luas min  7    16    16  6
cm
7
7
7
Contoh 3: hal. 357
Sebuah tabung (radius r2, tinggi h2) dimasukkan
kedalam kerucut (radius r1, tinggi h1).
Tentukan vol. maks tabung itu.
Jawab:
h1 – h2
Pakai perbandingan tangent :
h1 – h 2
r2
h1
h2
r2

h1  h 2
r1

h1
r2 . h1  r1 . h1  r1 . h 2
r2
h2
h2 
r1
r1 . h1  r2 . h1
r1
2
Vol tab ung  π r 2 h 2
r1
substitusi kan h 2 lalu cari
dV
dr 2
0
PROBLEMS from internet
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 ?
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.
7. Find the point (x, y) on the graph of y 
x
nearest the point (4, 0).
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 samakaki
11. 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?
20. A rectangular field of fixed area is to be enclosed and
divided into three lots by parallels to one of the sides.
What should be the relative dimensions of the field to
make the amount of fencing minimum?
21. A box is to be made of a piece of
cardboard 9 cm square by cutting equal
squares out of the corners and turning up
the sides. Find the volume of the largest
box that can be made in this way.
22. Find the volume of the largest box
that can be made by cutting equal
squares out of the corners of a piece
of cardboard of dimensions 15 x 24
cm, and then turning up the sides.
23. The perimeter of an isosceles triangle is p cm. Find its maximum area.
24. Find the most economical proportions for
a box with an open top and a square base.
25. Find the dimension of the largest
rectangular building that can be
placed on a right-triangular lot, facing
one of the perpendicular sides.
26. A lot has the form of a right triangle, with
perpendicular sides 60 and 80 cm long.
Find the length and width of the largest
rectangular building that can be erected,
facing the hypotenuse of the triangle.
27. A page is to contain 24 cm2 of print.
The margins at top and bottom are
1.5 cm, at the sides 1 cm.
Find the most economical
dimensions of the page.
28. Two posts, one 8 feet high and the
other 12 feet high, stand 15 ft apart.
They are to be supported by wires
attached to a single stake at ground
level. The wires running to the tops
of the posts. Where should the
stake be placed, to use the least
amount of wire?
29. A ship lies 6 miles from shore, and
opposite a point 10 miles farther
along the shore another
ship lies 18 miles offshore.
A boat from the first ship is to land
a passenger and then proceed
to the other ship. What is the least
distance the boat can travel?
30. Find the point on the curve: a2 y = x3
that is nearest the point (4a, 0).
31. Find the shortest distance
from the point (5, 0) to the
curve 2y2 = x3
32. Inscribe a circular cylinder of
maximum convex surface area
in a given circular cone.
33. Find the circular cone
of maximum volume
inscribed in a sphere
of radius a.
22. Find the circular cone
of minimum volume
circumscribed about a
sphere of radius a.