Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a…

Question

Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a fare of Ksh (2n+20) and the total fare collected was Ksh. 120,000. Determine the number of passengers. Locate the centre of enlargement and determine the linear scale factor.

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Haya hapa ni masuluhisho ya maswali yako:

1. Tatua $9x^2 + 16 = 24x$ .

Panga upya mlinganyo katika mfumo wa kawaida wa mlinganyo wa quadratic, $ax^2 + bx + c = 0$ .
$9x^2 - 24x + 16 = 0$
Tambua kwamba huu ni trinomial kamili ya mraba.
$(3x)^2 - 2(3x)(4) + (4)^2 = 0$ $(3x - 4)^2 = 0$
Tatua kwa $x$ .
$3x - 4 = 0$ $3x = 4$ $x = \frac{4}{3}$
Jibu la mwisho: $\boxed{x = \frac{4}{3}}$

2. Treni ya abiria ina abiria $n$ katika safari fulani. Kila abiria alilipa nauli ya Ksh $(2n+20)$ na jumla ya nauli iliyokusanywa ilikuwa Ksh 120,000. Bainisha idadi ya abiria.
Unda mlinganyo unaowakilisha jumla ya nauli iliyokusanywa.
Jumla ya nauli = (Idadi ya abiria) $\times$ (Nauli kwa kila abiria)
$120,000 = n(2n + 20)$
Panua na panga upya mlinganyo katika mfumo wa kawaida wa quadratic.
$120,000 = 2n^2 + 20n$ $2n^2 + 20n - 120,000 = 0$
Rahisisha mlinganyo kwa kugawanya pande zote mbili kwa 2.
$n^2 + 10n - 60,000 = 0$
Tumia fomula ya quadratic, $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , kutatua kwa $n$ .
Hapa, $a=1$ , $b=10$ , $c=-60,000$ .
$n = \frac{-10 \pm \sqrt{10^2 - 4(1)(-60,000)}}{2(1)}$ $n = \frac{-10 \pm \sqrt{100 + 240,000}}{2}$ $n = \frac{-10 \pm \sqrt{240,100}}{2}$ $n = \frac{-10 \pm 490}{2}$
Kokotoa thamani zinazowezekana za $n$ .
$n_1 = \frac{-10 + 490}{2} = \frac{480}{2} = 240$ $n_2 = \frac{-10 - 490}{2} = \frac{-500}{2} = -250$
Kwa kuwa idadi ya abiria haiwezi kuwa hasi, tunachukua thamani chanya.
Jibu la mwisho: Idadi ya abiria ni $\boxed{240}$ .

3. Tafuta kituo cha ukuzaji na bainisha kipimo cha ukuzaji (linear scale factor).
Tafuta kituo cha ukuzaji.
Kituo cha ukuzaji hupatikana kwa kuchora mistari iliyonyooka inayounganisha ncha zinazolingana za umbo asili ( $QRS$ ) na umbo lililokuzwa ( $Q'R'S'$ ). Kwa mfano, unganisha $Q$ na $Q'$ , $R$ na $R'$ , na $S$ na $S'$ . Sehemu ambapo mistari hii inakutana ndio kituo cha ukuzaji. Kutoka kwenye mchoro, sehemu hii imeandikwa kama $O$ .
Kituo cha ukuzaji ni O.
Bainisha kipimo cha ukuzaji (linear scale factor).
Kipimo cha ukuzaji ( $k$ ) ni uwiano wa urefu wa upande wa umbo lililokuzwa kwa urefu wa upande unaolingana wa umbo asili. Kutokana na mahesabu yaliyotolewa, tunatumia uwiano wa 8.7 kwa 3.
$k = \frac{\text{Urefu wa upande wa umbo lililokuzwa}}{\text{Urefu wa upande wa umbo asili}}$ $k = \frac{8.7}{3}$ $k = 2.9$
Jibu la mwisho: Kituo cha ukuzaji ni $\boxed{O}$ na kipimo cha ukuzaji ni $\boxed{2.9}$ .

Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a fare of Ksh (2n+20) and the total fare collected was Ksh. 120,000. Determine the number of passengers. Locate the centre of enlargement and determine the linear scale factor.

Step-by-step solution

Panga upya mlinganyo katika mfumo wa kawaida wa mlinganyo wa quadratic, $ax^2 + bx + c = 0$ .

Tambua kwamba huu ni trinomial kamili ya mraba.

Tatua kwa $x$ .

Unda mlinganyo unaowakilisha jumla ya nauli iliyokusanywa.

Panua na panga upya mlinganyo katika mfumo wa kawaida wa quadratic.

Rahisisha mlinganyo kwa kugawanya pande zote mbili kwa 2.

Tumia fomula ya quadratic, $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , kutatua kwa $n$ .

Kokotoa thamani zinazowezekana za $n$ .

Tafuta kituo cha ukuzaji.

Bainisha kipimo cha ukuzaji (linear scale factor).

Still stuck on this one?

More Mathematics Questions

Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a fare of Ksh (2n+20) and the total fare collected was Ksh. 120,000. Determine the number of passengers. Locate the centre of enlargement and determine the linear scale factor.

Step-by-step solution

Panga upya mlinganyo katika mfumo wa kawaida wa mlinganyo wa quadratic, ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Tambua kwamba huu ni trinomial kamili ya mraba.

Tatua kwa xxx.

Unda mlinganyo unaowakilisha jumla ya nauli iliyokusanywa.

Panua na panga upya mlinganyo katika mfumo wa kawaida wa quadratic.

Rahisisha mlinganyo kwa kugawanya pande zote mbili kwa 2.

Tumia fomula ya quadratic, n=−b±b2−4ac2an = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}n=2a−b±b2−4ac​​, kutatua kwa nnn.

Kokotoa thamani zinazowezekana za nnn.

Tafuta kituo cha ukuzaji.

Bainisha kipimo cha ukuzaji (linear scale factor).

Still stuck on this one?

More Mathematics Questions

Panga upya mlinganyo katika mfumo wa kawaida wa mlinganyo wa quadratic, $ax^2 + bx + c = 0$ .

Tatua kwa $x$ .

Tumia fomula ya quadratic, $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , kutatua kwa $n$ .

Kokotoa thamani zinazowezekana za $n$ .