The figure below shows a net of a prism whose cross - section is an equilateral triangle. Sketch the prism. Solve for x in the equation 32^(x-3) 8^(x-4) = 64 2^x

Huu hapa ni utatuzi wa maswali:

7. Sketch the prism Prism yenye sehemu-mtambuka ya pembetatu pacha (equilateral triangle) inaitwa triangular prism. Inaundwa na nyuso mbili za pembetatu pacha (kama besi) na nyuso tatu za mstatili (kama pande).

Ili kuchora prism, fuata hatua hizi:

1. Chora pembetatu pacha moja kwa ajili ya uso wa mbele.
2. Kutoka kila kona ya pembetatu ya mbele, chora mistari mitatu inayofanana na yenye urefu sawa kuelekea nyuma na chini (kuonyesha kina).
3. Unganisha ncha za mistari hii mitatu kwa mistari mingine mitatu ili kuunda pembetatu ya pili inayofanana na ya kwanza, ambayo itakuwa uso wa nyuma.
4. Mistari ya nyuma ambayo haionekani inaweza kuchorwa kama mistari iliyokatika (dashed lines).

Hapa kuna maelezo ya jinsi inavyoonekana:

• Ina besi mbili za pembetatu pacha.
• Ina nyuso tatu za mstatili zinazounganisha besi hizo.
• Nyuso zote za mstatili zitakuwa na urefu sawa na urefu wa prism, na upana sawa na urefu wa upande wa pembetatu pacha.

Solve for x in the equation Tunatakiwa kutatua mlinganyo $32^{(x-3)} \div 8^{(x-4)} = 64 \div 2^x$.

Step 1: Andika nambari zote kama nguvu za 2. Tunajua kuwa $32 = 2^5$, $8 = 2^3$, na $64 = 2^6$. Badilisha hizi kwenye mlinganyo: $(2^5)^{(x-3)} \div (2^3)^{(x-4)} = 2^6 \div 2^x$

Step 2: Tumia sheria ya nguvu $(a^m)^n = a^{mn}$. $2^{5(x-3)} \div 2^{3(x-4)} = 2^6 \div 2^x$ $2^{5x-15} \div 2^{3x-12} = 2^6 \div 2^x$

Step 3: Tumia sheria ya mgawanyiko wa nguvu $a^m \div a^n = a^{m-n}$. $2^{(5x-15) - (3x-12)} = 2^{6-x}$ $2^{5x-15 - 3x + 12} = 2^{6-x}$ $2^{2x-3} = 2^{6-x}$

Step 4: Linganisha viashiria (exponents) kwa sababu besi ni sawa. $2x - 3 = 6 - x$

Step 5: Tatua kwa $x$. Ongeza $x$ pande zote mbili: $2x + x - 3 = 6$ $3x - 3 = 6$ Ongeza 3 pande zote mbili: $3x = 6 + 3$ $3x = 9$ Gawanya kwa 3: $x = \frac{9}{3}$ $x = 3$

Jibu la mwisho ni: $\boxed{x = 3}$ 3 done, 2 left today. You're making progress.