Bu'aan kutaa 12ffaa ba'eera; bu'aa kee eessatti akka ilaaltu

Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 4 Herrega kutaa 12: Seensa Liiniyar Piroogiraamiingii

Wanti fuula kana irra jiru hundi boqonnaa 4 kitaaba Herrega kutaa 12 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

Google Play irraa buufadhaa
4
Kaardii
10
Gaaffii shaakalaa boqonnaa 4
237–302
Fuula kitaabaa boqonnaa 4

Boqonnaan 4 maal haguuga

  1. 1.Graphical Solutions of System of Linear Inequalities
  2. 2.Maximum and Minimum Values
  3. 3.Applications

Yaadannoo gabaabaa boqonnaa 4

Kaardiiwwan fuula Herrega kutaa 12 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Akaakuu Qixxaattoo-dhabeeyyiiMallattooleeSarara DaangaaQabxiileen Sarararraa Ni Dabalamu
Haphina<,><, >Sarara citiinsa qabuLakki
Laafaa≤,≥\le, \geSarara guutuuEeyyee

Yoom akka itti fayyadamtu

Akaakuu sararaa sirrii filachuu fi sararri daangaa naannoo furmaataa keessatti qabamuu isaa murteessuuf gabatee kanatti fayyadamaa.

Of eeggadhu

Qixxaattoo-dhabeeyyii haphinaatiif sarara guutuu sararuun yookiin isa laafaaf sarara citiinsa qabu sararuun qabxii nama dhabsiisa.

Grade 12 Mathematics, fuula 237-302 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Yoom akka itti fayyadamtu

Yeroo gama lamaan qixxaattoo-dhabeeyyii lakkoofsa negatiiviin baay'istan yookiin hirtan seerri kun hojiirra oola.

Of eeggadhu

Lakkoofsa negatiiviin baay'isuun yookiin hiruun mallattoo qixxaattoo-dhabeeyyii ni garagalcha, garuu lakkoofsa kamiyyuu ida'uun yookiin hir'isuun mallatticha hin jijjiiru.

Grade 12 Mathematics, fuula 237-302 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Yoom akka itti fayyadamtu

Qixxaattoo-dhabeeyyii sararaawaa jijjiiramaa lamaatiif gama sarara daangaa isa kam akka hallulleessitan murteessuuf tooftaa kanatti fayyadamaa.

Of eeggadhu

Qabxii qormaataa sarara daangaa irratti argamu matumaa hin filatinaa. Sararichi tuqaa jalqabaa keessa yoo darbe (0, 0) bakka buusuun qabxii akka (0, 1) yookiin (1, 0) qoraa.

Grade 12 Mathematics, fuula 237-302 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Yoom akka itti fayyadamtu

Gaaffilee jechootaa fi daangaawwan qabatamaa gara qixxaattoo-dhabeeyyii sararaawaatti yeroo jijjiirtan fayyadamaa.

Of eeggadhu

Yoo baay'ate jechuun xiqqaataa yookiin qixxee gara jedhutti hiikama malee guddataa jechuu miti. Yoo xinnaate jechuun immoo guddataa yookiin qixxee gara jedhutti hiikama.

Grade 12 Mathematics, fuula 237-302 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 4

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Solve the inequality 7x + 3 < 5x + 9.

  1. A.x &gt; 3
  2. B.x &lt; 3
  3. C.x &lt; 6
  4. D.x &gt; 6
Deebii fi sababa agarsiisi

Deebii: B x &lt; 3

Sababa

Subtract 5x from both sides to get 2x + 3 < 9. Then subtract 3 from both sides to get 2x < 6. Divide both sides by the positive number 2 to get x < 3. The sign stays the same because you did not divide by a negative number.

Gaaffii 2

Alem has Birr 500 to buy wheat flour. One packet costs Birr 130. If x is the number of packets, the inequality is 130x < 500. Which value of x is not a solution?

  1. A.0
  2. B.1
  3. C.3
  4. D.4
Deebii fi sababa agarsiisi

Deebii: D 4

Sababa

When x = 4, 130 × 4 = 520. But 520 is not less than 500, so the statement is false. The values 0, 1, 2 and 3 all make the inequality true.

Gaaffii 3

In the inequality y < 5x + 7, the boundary line is drawn as a dashed (broken) line. Why?

  1. A.The slope of the line is 7.
  2. B.The line is vertical.
  3. C.The points on the line are not included, because the sign is strict (&lt;).
  4. D.The point (0, 0) does not satisfy the inequality.
Deebii fi sababa agarsiisi

Deebii: C The points on the line are not included, because the sign is strict (&lt;).

Sababa

When the sign is < or >, the points on the line are not part of the solution. So you draw a dashed line. When the sign is ≤ or ≥, the points on the line are included, and you draw a solid line.

Gaaffii 4

A taxi in Addis Ababa charges the total cost T = 0.75x + 1.75, where x is the distance in km and T is in Birr. Selam has at most Birr 10 to spend. What is the maximum distance she can travel?

  1. A.8.25 km
  2. B.10 km
  3. C.11 km
  4. D.12 km
Deebii fi sababa agarsiisi

Deebii: C 11 km

Sababa

Write 0.75x + 1.75 < 10. Subtract 1.75 to get 0.75x < 8.25. Divide by 0.75 to get x < 11. So the maximum distance is 11 km.

Gaaffii 5

A furniture dealer wants to maximize profit. The objective function is Z = 300x + 100y, where x is the number of tables and y is the number of chairs. The corner points of the feasible region are (0, 0), (20, 0), (10, 50) and (0, 60). What is the maximum profit?

  1. A.Birr 3,000
  2. B.Birr 6,000
  3. C.Birr 8,000
  4. D.Birr 0
Deebii fi sababa agarsiisi

Deebii: C Birr 8,000

Sababa

Substitute each corner point into Z. At (0, 0), Z = 0. At (20, 0), Z = 6000. At (10, 50), Z = 3000 + 5000 = 8000. At (0, 60), Z = 6000. The largest value is 8000, at (10, 50), which means 10 tables and 50 chairs.

Gaaffii 6

The dealer has Birr 50,000 to invest. A table costs Birr 2500 and a chair costs Birr 500. Which simplified inequality shows this investment limit? (x = number of tables, y = number of chairs)

  1. A.x + 5y &le; 100
  2. B.5x + y &le; 100
  3. C.5x + y &ge; 100
  4. D.2500x + 500y &ge; 50000
Deebii fi sababa agarsiisi

Deebii: B 5x + y &le; 100

Sababa

The total spent is 2500x + 500y, and it must be at most 50000. Divide every term by 500 to get 5x + y ≤ 100.

Gaaffii 7

A linear programming problem has the objective function Z = x + y. The feasible region has corner points (0, 0), (1, 0), (0, 2) and (2, 1). At which point is Z largest?

  1. A.(0, 0)
  2. B.(1, 0)
  3. C.(0, 2)
  4. D.(2, 1)
Deebii fi sababa agarsiisi

Deebii: D (2, 1)

Sababa

Calculate Z at each corner. At (0, 0), Z = 0. At (1, 0), Z = 1. At (0, 2), Z = 2. At (2, 1), Z = 3. The largest value is 3, at (2, 1).

Gaaffii 8

Ficho's budget constraints are: the number of answer sheets (x) is at least 40 more than the number of pencils (y), and the cost is x + 2y ≤ 400 (sheets cost Birr 1 each, pencils cost Birr 2 each). Can Ficho buy 150 answer sheets and 100 pencils?

  1. A.No, because the sheets cost more than the pencils.
  2. B.No, because the total cost is more than Birr 400.
  3. C.Yes, because both conditions are met.
  4. D.Yes, but only if the number of sheets equals the number of pencils.
Deebii fi sababa agarsiisi

Deebii: C Yes, because both conditions are met.

Sababa

Check the first condition: 150 is at least 100 + 40 = 140, so it holds. Check the cost: 150 + 2 × 100 = 350, which is at most 400, so it holds. Both conditions are met, so the purchase is possible.

Gaaffii 9

A world feed programme needs at least 800 kg of special feed each day. The minimum cost objective is Z = 30x + 90y, where x is kg of corn and y is kg of soybean meal. The two corner points that are candidates for the optimum are (400, 400) and (480, 320). What is the minimum daily cost?

  1. A.Birr 43,200
  2. B.Birr 48,000
  3. C.Birr 30,000
  4. D.Birr 90,000
Deebii fi sababa agarsiisi

Deebii: A Birr 43,200

Sababa

At (400, 400), Z = 30 × 400 + 90 × 400 = 48,000. At (480, 320), Z = 30 × 480 + 90 × 320 = 14,400 + 28,800 = 43,200. The smaller value is the minimum cost, which is Birr 43,200 per day.

Gaaffii 10

A problem asks to minimize Z = 2x + 4y. The corner points are (0, 10), (2, 4) and (10, 0), giving Z values of 40, 20 and 20. Which statement is correct?

  1. A.The minimum is reached only at (2, 4).
  2. B.The minimum value is 20, and every point on the segment between (2, 4) and (10, 0) gives this value, so the solution is not unique.
  3. C.The maximum value of Z is 40, so the minimum is 40.
  4. D.The minimum value is 40, at the point (0, 10).
Deebii fi sababa agarsiisi

Deebii: B The minimum value is 20, and every point on the segment between (2, 4) and (10, 0) gives this value, so the solution is not unique.

Sababa

Both (2, 4) and (10, 0) give Z = 20, which is the smallest value at the corners. Every point on the line segment between them also gives Z = 20. So the optimal solution is not unique. The value 40 at (0, 10) is larger, not smaller.

Appiin boqonnaa kana irratti gaaffii 12 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.

Fuula kitaabaa boqonnaa 4

Boqonnaan 4 Maxxansa Ministeera Barnootaa keessatti fuula 237 hanga fuula 302 ti.

Maxxansa Ministeera Barnootaa, Ingiliffaa, Fuula 426

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Yaadannoo gabaabaa Herrega kutaa 12 hunda

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