Question

22. Show thar 3200 is not a perfect square number.
23. Hew many pencil boxes of dimensions 15cm x 50cm x 20cm can be packed in a carton each of
whose edge is 30cm?
24. The price of an article is reduced by 20%. find the original price of the article, if the reduced price is
Rs 4000
25. The length of the parallel sides of a trapezium are 12 cm and 15 cm respectively and the distance
berween the parallel sides is 8 cm then find the area of the trapezium.
20. Find the value of x in 16^3x=32^5x-13​

Answer 1

$\large\underline{\sf{Solution-22}}$

$\sf \: Prime \: factorization \: of \: 3200 \: is \:$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:3200\: \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:1600\:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:800\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:400\:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:200\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:100\:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:50\:\:}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:25\:\:}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}} \\{\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\: \: \: \: \: \: \: \bull \: \sf \: 3200 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5$

$\rm :\implies\:3200 = {2}^{2} \times {2}^{2} \times {2}^{2} \times {5}^{2} \times 2$

• Since, we have one unpaired 2 left, so 3200 is not a perfect square.

$\large\underline{\sf{Solution-23}}$

Dimensions of Pencil Box

• Length of pencil box, l = 15 cm

• Breadth of pencil box, b = 50 cm

• Height of pencil box, h = 20 cm

It implies,

• Pencil box is cuboidal in shape,

$\rm :\longmapsto\:Volume_{(pencil \: box)} = l \times b \times h$

$\rm :\longmapsto\:Volume_{(pencil \: box)} = 15 \times 50 \times 20 = 15000 \: {cm}^{3}$

Now,

• Carton is in the shape of cube, whose side is 30 cm.

Therefore,

$\rm :\longmapsto\:Volume_{(carton)} = {(edge)}^{3}$

$\rm :\longmapsto\:Volume_{(carton)} = {(30)}^{3} = 27000 \: {cm}^{3}$

So,

$\rm :\longmapsto\:Number_{(pencilbox)} = \dfrac{Volume_{(carton)}}{Volume_{(pencilbox)}}$

$\rm :\longmapsto\:Number_{(pencilbox)} =\dfrac{27000}{15000} = 1.8$

• It implies, only 1 pencil box is packed in this carton.

$\large\underline{\sf{Solution-24}}$

$\sf \: Let \: the \: original \: price \: be \: Rs \: x.$

Now,

• The price reduced by 20 %

and

• reduced price is Rs 4000.

$\rm :\implies\:x - 20\% \: of \: x \: = \: 4000$

$\rm :\longmapsto\:x - \dfrac{20x}{100} = 4000$

$\rm :\longmapsto\:x - \dfrac{x}{5} = 4000$

$\rm :\longmapsto\:\dfrac{5x - x}{5} = 4000$

$\rm :\longmapsto\:4x = 20000$

$\bf\implies \:x \: = \: Rs \: 5000$

$\bf\implies \:The \: original \: price \: of \: article \: is \: Rs \: 5000$

$\large\underline{\sf{Solution-25}}$

Dimensions of trapezium are

• The length of the parallel sides of a trapezium are 12 cm and 15 cm respectively.

• The distance between the parallel sides is 8 cm.

$\rm :\longmapsto\:Area_{(trapezium)} = \dfrac{1}{2} (sum \: of \: \parallel \: sides) \times distance$

$\rm :\longmapsto\:Area_{(trapezium)} = \dfrac{1}{2} (12 + 15) \times 8$

$\bf :\longmapsto\:Area_{(trapezium)} = 27 \times 4 = 108 \: {cm}^{2}$

$\large\underline{\sf{Solution-20}}$

$\rm :\longmapsto\: {16}^{3x} = {32}^{5x - 13}$

$\rm :\longmapsto\: {2}^{4 \times 3x} = {2}^{5 \times (5x - 13)}$

On comparing, we get

$\rm :\longmapsto\:12x = 5 \times (5x - 13)$

$\rm :\longmapsto\:12x = 25x - 65$

$\rm :\longmapsto\:12x - 25x = - 65$

$\rm :\longmapsto\: - 13x = - 65$

$\bf\implies \:x = 5$