Your sister work as a production manager in The company that deals in manufacturing of school laboratory supplies has received an order of making 1420 pyramids for a certain school. The school wants a pyramid with the square base measuring 6cm and slanting edge of 7cm. Your sister must submit a sketch that to be folded to make a pyramid that the school wants but she is finding it hard. The company packs the pyramid in a box that measures 66cm by 24cm by 15cm and the manager wants to know the number of boxes that will be needed to pack all the pyramids that the school wants as they are only 5 boxes Available in the stalk. The company manufactures and sells each pyramid at shs. 25000 and the company manager receives a commission of 12% on the first 500 pyramids. Then 8%on the pyramids excess of 500. Your sister wants to know the total commission she will receive from the order made by the school. Task: a) help your sister to sketch the diagram she is to submit. b) how many pyramids can be packed in the box (volume). c) are the available boxes enough for the pyramids ordered for. d) obtain the total commission that your sister will get from the order

Question 1

Answer · 2026-06-18T23:28:30.403Z

a) Mchoro wa piramidi Ili kutengeneza piramidi yenye msingi wa mraba na kingo za pembeni, mchoro (net) utakuwa na sehemu zifuatazo: • Msingi wa mraba: Mraba mmoja wenye urefu wa upande wa 6 cm. • Nyuso za pembeni za pembetatu: Pembetatu nne zilizounganishwa kwenye kila upande wa mraba. Kila pembetatu itakuwa na msingi wa 6 cm (upande wa mraba) na kingo mbili za pembeni zenye urefu wa 7 cm kila moja (kingo za piramidi). b) Idadi ya piramidi zinazoweza kuwekwa kwenye sanduku (kwa kutumia ujazo) Step 1: Tafuta urefu wa piramidi (h). Kwanza, tunahitaji urefu wa pembeni wa pembetatu (slant height, l). l² + (upande wa msingi/2)² = (kingo ya pembeni)² l² + (6/2)² = 7² l² + 3² = 49 l² + 9 = 49 l² = 40 l = √40 cm Sasa, tafuta urefu wa piramidi (h). h² + (upande wa msingi/2)² = l² h² + 3² = (√40)² h² + 9 = 40 h² = 31 h = √31 cm ≈ 5.568 cm Step 2: Kokotoa ujazo wa piramidi moja. Ujazo wa piramidi = (1/3) × Eneo la Msingi × Urefu Eneo la Msingi = upande² = (6 cm)² = 36 cm² Ujazo wa piramidi = (1/3) × 36 cm² × √31 cm Ujazo wa piramidi = 12√31 cm³ ≈ 12 × 5.568 cm³ ≈ 66.816 cm³ Step 3: Kokotoa ujazo wa sanduku. Ujazo wa sanduku = Urefu × Upana × Kimo Ujazo wa sanduku = 66 cm × 24 cm × 15 cm Ujazo wa sanduku = 23760 cm³ Step 4: Kokotoa idadi ya piramidi zinazoweza kuwekwa kwenye sanduku. Idadi ya piramidi = Ujazo wa sanduku / Ujazo wa piramidi moja Idadi ya piramidi = 23760 cm³ / (12√31 cm³) Idadi ya piramidi = 1980 / √31 ≈ 355.60 Kwa kuwa huwezi kuweka sehemu ya piramidi, tunachukua nambari kamili. Idadi ya piramidi zinazoweza kuwekwa kwenye sanduku ni 355. c) Je, masanduku yaliyopo yanatosha kwa piramidi zilizoagizwa? Step 1: Kokotoa idadi ya masanduku yanayohitajika. Jumla ya piramidi zilizoagizwa = 1420 Idadi ya piramidi kwa kila sanduku = 355 (kutoka sehemu b) Idadi ya masanduku yanayohitajika = 1420 / 355 = 4 masanduku Step 2: Linganisha na masanduku yaliyopo. Masanduku yaliyopo = 5 Kwa kuwa masanduku 4 yanahitajika na masanduku 5 yanapatikana, ndiyo, masanduku yaliyopo yanatosha kwa piramidi zilizoagizwa. d) Jumla ya kamisheni ambayo dada yako atapata kutoka kwa agizo hilo. Step 1: Kokotoa kamisheni kwa piramidi 500 za kwanza. Bei ya piramidi moja = Shs. 25000 Kamisheni ya 12% kwa piramidi 500 za kwanza: Mapato ya piramidi 500 = 500 × Shs. 25000 = Shs. 12,500,000 Kamisheni = 12% ya Shs. 12,500,000 = 0.12 × Shs. 12,500,000 = Shs. 1,500,000 Step 2: Kokotoa kamisheni kwa piramidi zinazozidi 500. Idadi ya piramidi zinazozidi 500 = 1420 - 500 = 920 piramidi Kamisheni ya 8% kwa piramidi 920 zinazozidi: Mapato ya piramidi 920 = 920 × Shs. 25000 = Shs. 23,000,000 Kamisheni = 8% ya Shs. 23,000,000 = 0.08 × Shs. 23,000,000 = Shs. 1,840,000 Step 3: Kokotoa jumla ya kamisheni. Jumla ya kamisheni = Kamisheni ya kwanza + Kamisheni ya ziada Jumla ya kamisheni = Shs. 1,500,000 + Shs. 1,840,000 Jumla ya kamisheni ambayo dada yako atapata ni Shs. 3,340,000.

Question 2

a) Mchoro wa piramidi Ili kutengeneza piramidi yenye msingi wa mraba na kingo za pembeni, mchoro (net) utakuwa na sehemu zifuatazo: • Msingi wa mraba: Mraba mmoja wenye urefu wa upande wa 6 cm. • Nyuso za pembeni za pembetatu: Pembetatu nne zilizounganishwa kwenye kila upande wa mraba. Kila pembetatu itakuwa na msingi wa 6 cm (upande wa mraba) na kingo mbili za pembeni zenye urefu wa 7 cm kila moja (kingo za piramidi). b) Idadi ya piramidi zinazoweza kuwekwa kwenye sanduku (kwa kutumia ujazo) Step 1: Tafuta urefu wa piramidi (h). Kwanza, tunahitaji urefu wa pembeni wa pembetatu (slant height, l). l² + (upande wa msingi/2)² = (kingo ya pembeni)² l² + (6/2)² = 7² l² + 3² = 49 l² + 9 = 49 l² = 40 l = √40 cm Sasa, tafuta urefu wa piramidi (h). h² + (upande wa msingi/2)² = l² h² + 3² = (√40)² h² + 9 = 40 h² = 31 h = √31 cm ≈ 5.568 cm Step 2: Kokotoa ujazo wa piramidi moja. Ujazo wa piramidi = (1/3) × Eneo la Msingi × Urefu Eneo la Msingi = upande² = (6 cm)² = 36 cm² Ujazo wa piramidi = (1/3) × 36 cm² × √31 cm Ujazo wa piramidi = 12√31 cm³ ≈ 12 × 5.568 cm³ ≈ 66.816 cm³ Step 3: Kokotoa ujazo wa sanduku. Ujazo wa sanduku = Urefu × Upana × Kimo Ujazo wa sanduku = 66 cm × 24 cm × 15 cm Ujazo wa sanduku = 23760 cm³ Step 4: Kokotoa idadi ya piramidi zinazoweza kuwekwa kwenye sanduku. Idadi ya piramidi = Ujazo wa sanduku / Ujazo wa piramidi moja Idadi ya piramidi = 23760 cm³ / (12√31 cm³) Idadi ya piramidi = 1980 / √31 ≈ 355.60 Kwa kuwa huwezi kuweka sehemu ya piramidi, tunachukua nambari kamili. Idadi ya piramidi zinazoweza kuwekwa kwenye sanduku ni 355. c) Je, masanduku yaliyopo yanatosha kwa piramidi zilizoagizwa? Step 1: Kokotoa idadi ya masanduku yanayohitajika. Jumla ya piramidi zilizoagizwa = 1420 Idadi ya piramidi kwa kila sanduku = 355 (kutoka sehemu b) Idadi ya masanduku yanayohitajika = 1420 / 355 = 4 masanduku Step 2: Linganisha na masanduku yaliyopo. Masanduku yaliyopo = 5 Kwa kuwa masanduku 4 yanahitajika na masanduku 5 yanapatikana, ndiyo, masanduku yaliyopo yanatosha kwa piramidi zilizoagizwa. d) Jumla ya kamisheni ambayo dada yako atapata kutoka kwa agizo hilo. Step 1: Kokotoa kamisheni kwa piramidi 500 za kwanza. Bei ya piramidi moja = Shs. 25000 Kamisheni ya 12% kwa piramidi 500 za kwanza: Mapato ya piramidi 500 = 500 × Shs. 25000 = Shs. 12,500,000 Kamisheni = 12% ya Shs. 12,500,000 = 0.12 × Shs. 12,500,000 = Shs. 1,500,000 Step 2: Kokotoa kamisheni kwa piramidi zinazozidi 500. Idadi ya piramidi zinazozidi 500 = 1420 - 500 = 920 piramidi Kamisheni ya 8% kwa piramidi 920 zinazozidi: Mapato ya piramidi 920 = 920 × Shs. 25000 = Shs. 23,000,000 Kamisheni = 8% ya Shs. 23,000,000 = 0.08 × Shs. 23,000,000 = Shs. 1,840,000 Step 3: Kokotoa jumla ya kamisheni. Jumla ya kamisheni = Kamisheni ya kwanza + Kamisheni ya ziada Jumla ya kamisheni = Shs. 1,500,000 + Shs. 1,840,000 Jumla ya kamisheni ambayo dada yako atapata ni Shs. 3,340,000.

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