Feàrrachadh - Eisimpleir
'S e an tomhas-lìonaidh aig a' chiùbaid gu h-ìosal 72 cm3.
- Seall gur e \(F(x) = 12\left( {{x^2} + \frac{{10}}{x}} \right)\) farsaingeachd-uachdair na ciùbaid
- Obraich a-mach an fharsaingeachd-uachdair as lugha
Fuasgladh
1. \({F_{Aghaidh}} = faid \times leud\)
\(= 3x \times \text{àirde}\)
\(= 3x\text{à}\,aonada{n^2}\)
\({F_{\text{Cùl}}} = faid \times leud\)
\(= 3x \times \text{à}\)
\(= 3x\text{à}\,aonada{n^2}\)
\({F_{\text{Clì}}} = faid \times leud\)
\(= 2x \times \text{à}\)
\(= 2x\text{à}\,aonada{n^2}\)
\({F_{Deas}} = faid \times leud\)
\(= 2x \times \text{à}\)
\(= 2x\text{à}\,aonada{n^2}\)
\({F_{Mullach}} = faid \times leud\)
\(= 3x \times 2x\)
\(= 6{x^2}\,aonada{n^2}\)
\({F_{Bonn}} = faid \times leud\)
\(= 3x \times 2x\)
\(= 6{x^2}\,aonada{n^2}\)
Farsaingeachd-uachdair iomlan \(= 10xh + 12{x^2}\)
Chan eil seo fhathast a' maidseadh na th' againn ri dhearbhadh oir tha an teirm \(\text{à}\) ann, agus tha sinn ag iarraidh abairt ann an teirmean de \(x\) a-mhàin.
Chì sinn bho thoiseach na ceist gur e 72 cm3 tomhas-lìonaidh na ciùbaid.
\(Tomas-lionaid{h_{ciubaid}} = faid \times leud \times \text{àirde}\)
\(= 3x \times 2x \times \text{à}\)
\(= 6{x^2}\text{à}\)
Mar sin \(6{x^2}\text{à} = 72\)
\(\text{à} = \frac{{72}}{{6{x^2}}}\)
\(\text{à} = \frac{{12}}{{{x^2}}}\)
Faodaidh sinn a-nis seo ionadachadh a-steach dhan abairt againn airson an fharsaingeachd-uachdair iomlan obrachadh a-mach.
\(F(x) = 10x \times \frac{{12}}{{{x^2}}} + 12{x^2}\)
\(F(x) = 12{x^2} + \frac{{120}}{x}\)
\(F(x) = 12x\left( {x + \frac{{10}}{x}} \right)\,aonada{n^2}\) mar a bhios a dhìth.
2. A-rithist, gus an fharsaingeachd-uachdair as lugha obrachadh a-mach, feumaidh sinn faighinn a-mach dè an luach de \(x\) a bheir dhuinn a' phuing-tionndaidh as ìsle agus an uair sin an fharsaingeachd obrachadh a-mach a rèir sin
\(F(x) = 12\left( {{x^2} + \frac{{10}}{x}} \right)\)
\(F(x) = 12{x^2} + \frac{{120}}{x}\)
\(F(x) = 12{x^2} + 120{x^{ - 1}}\)
\(F\textquotesingle (x) = 24x - 120{x^{ - 2}}\)
\(F\textquotesingle (x) = 24x - \frac{{120}}{{{x^2}}}\)
Bidh puingean neo-ghluasadach ann nuair a tha \(\frac{{dy}}{{dx}} = 0\).
\(24x - \frac{{120}}{{{x^2}}} = 0\)
Iomadaich gach teirm le \({x^2}\)
\(24{x^3} - 120 = 0\)
\(24{x^3} = 120\)
\({x^3} = \frac{{120}}{{24}}\)
\({x^3}=5\)
\(x= \sqrt[3]{5}\)
Nàdar
\(\frac{{dy}}{{dx}} = 24(1) - \frac{{120}}{{{1^2}}} = - 96\) (àicheil)
\(\frac{{dy}}{{dx}} = 24(\sqrt[3]{5}) - \frac{{120}}{{{{(\sqrt[3]{5})}^2}}} = 0\) (neo-ghluasadach)
\(\frac{{dy}}{{dx}} = 24(2) - \frac{{120}}{{{2^2}}} = 18\) (dearbhte)
Mar sin tha an fharsaingeachd-uachdair as lugha nuair a tha \(x = \sqrt[3]{5}\)
\(F(x) = 12\left( {{x^2} + \frac{{10}}{x}} \right)\)
\(F(\sqrt[3]{5}) = 12\left( {{{(\sqrt[3]{5})}^2} + \frac{{10}}{{\sqrt[3]{5}}}} \right)\)
\(F(\sqrt[3]{5}) = 105.3\,aonada{n^2}(gu\,1\,id.)\)
Mar sin 's e an fharsaingeachd-uachdair as lugha \(105.3\,aonada{n^2}\)
