Question

rk fabricators has got a order for making a frame for machine of their client. for which, they are using a autocad software to create a constructible model that includes the relevant information such as dimensions of the frame and materials needed. the frame will have a solid base and will be cut out of a piece of steel the final area of the frame should be 54 sq m. the digram of frame is shown below. in order to input the right values in the autocad software, the engineer needs to calculate some basic things. on the basis of the above information, answer any four of the following questions: (i) what are the dimensions of the outer frame ? (a) 10 + x and 5 + x (b) 10 - x and 5 - x (c) 10 + 2x and 5 + 2x (d) 10 - 2x and 5 – 2x(ii)a metal sheet of minimum area is used to make the frame. what should be the minimum area of metal sheet before cutting ? (a) 4x2 + 30x +50 (b) x2+ 27x +55 (c) 5x2+30 (d) 4x2 + 50 (iii)what is the area of required final metal frame ? (a) 4x2 + 30x +50 m2 (b) x2+ 27x +55 m2 (c) 4x2+50x m2 (d) 4x2 + 30 m2 (iv) if the area of the frame is 54 sq m, what is the value of x ? (a) 0.75 m (b) 3.0 m (c) 1.5 m (d) 1.8 m (v) what is the perimeter of the frame? (a) 36 m (b) 42 m (c) 45 m (d) 39 m​

Answer 1

Given : final area of the frame should be 54 sq m

Inner rectangle dimensions are l-10 b-5.

Outer is x meters from all sides  

To Find :  dimensions of the outer frame

Area of frame

Value of x

Perimeter of frame

Solution:

dimensions of the outer frame

10 + x + x   and 5 + x + x

= 10 + 2x  and 5 + 2x

Correct option is :   c) (10 +2x) and (5 + 2x)

Area of frame  =  (10 +2x)(5 + 2x)  = 50 + 20x  + 10x  + 4x²

= 4x²  + 30x + 50

area of required final metal frame

=  4x²  + 30x + 50 -  ( 10 * 5)

=   4x²  + 30x

4x²  + 30x  = 54

=> 2x²  +15x  = 27

=> 2x²  +15x  - 27 = 0

=> 2x² + 18x - 3x - 27 = 0

=> 2x(x + 9) - 3(x + 9) = 0

=> (2x - 3)(x + 9) = 0

=> x = 3/2    x  = - 9 ( x can not be negative )

Hence x = 1.5 m

Perimeter of frame :  2(10 + 2x + 5 + 2x)  = 30  + 8x

=30 + 8(1.5)

= 42 m

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Answer 2

Step-by-step explanation:

1. dimensions= (10+x+x) and (5+x+x)

(10+2x) and (5+2x)

2. minimum area= l x b

(10+2x)(5+2x)

50+20x+10x+4x²

4x²+30x+50

3. area of final frame= minimum area-area of

inner rectangle

4x²+30x+50 - 10 x 5

4x²+30x+50-50

4x²+30

4. value of x= 4x²+30x-54

2x²+15x-27

2x²+18x-3x+27

2x(x+9)-3(x+9)

2x-3=0 and x+9=0

x=3/2. and x=-9

value of x = 3/2= 1.5

5. perimeter of the frame= 2(10+2x+5+2x)

2(15+4(1.5))

30+12

42