Question

Only need help with parts (b) and (c)!! thank you!!

Answer 1

a) (i) 86.614 in

(ii) 47.6377 in

(iii) 1.584 m

b) width = 1.107 m (3 d.p.)

c) D = 1.135 m (3 d.p.)

Explanation:

Part (a)

Question (i)

Given:

• 1 m = 39.37 in

`\implies \sf 2.2\: m = 2.2 * 39.37 = 86.614\: in`

Question (ii)

Substitute the found value of D from part (i) into the given formula for D and solve for S:

`\sf \implies D=(S)/(0.55)`

`\sf \implies 86.614=(S)/(0.55)`

`\implies \sf S = 86.614 * 0.55`

`\implies \sf S = 47.6377\: in`

Question (iii)

Given:

• E = 1.1 m
• D = 2.2 m

Substitute the given values into the given formula for M and solve for M:

`\implies \sf M = E + D * 0.22`

`\implies \sf M = 1.1 + 2.2 * 0.22`

`\implies \sf M = 1.1 +0.484`

`\implies \sf M = 1.584 \: m`

Part (b)

Given:

• S = 50 in
• Ratio of width to height = 16 : 9

Pythagoras Theorem

`a^2+b^2=c^2`

where:

• a and b are the legs of the right triangle.
• c is the hypotenuse (longest side) of the right triangle.

Use Pythagoras Theorem to find the ratio of the diagonal S to the width and height:

`\implies \sf 16^2+9^2=S^2`

`\implies \sf 256+81=S^2`

`\implies \sf S^2=337`

`\implies \sf S=√(337)`

Therefore, the ratio of width to height to diagonal S of the TV is:

`\sf w:h:S=16 : 9 : √(337)`

If S = 50 in then:

`\implies \sf w:50=16:√(337)`

`\implies \sf (w)/(50)=(16)/(√(337))`

`\implies \sf w=(16)/(√(337)) * 50`

`\implies \sf width=43.57877686..\:in`

To convert to meters, divide the width in inches by 39.37:

`\implies \sf width=(43.57877686)/(39.37)=1.106903146\:m`

Therefore, the width of the TV is 1.107 m (3 d.p.).

Part (c)

If the TV is mounted in the position from part (aiii), where the midpoint of the TV is 1.584 m from the ground, and the height of the TV is 62.2cm, then:

`\implies \textsf{Top of TV} \sf = 1.584 + (62.2 / 100)/(2)`

`\implies \textsf{Top of TV} \sf = 1.895\:m`

Therefore, the top of the TV from the floor is 1.895m.

As E = 1.1 m, the vertical distance between the eye level and the top of the TV is:

`\implies \sf 1.895-1.1=0.795\:m`

To find D, model as a right triangle (see attached) and use the tan trigonometric ratio to find the shortest value of D.

Tan trigonometric ratio

`\sf \tan(\theta)=(O)/(A)`

where:

• 
  `\theta` is the angle
• O is the side opposite the angle
• A is the side adjacent the angle

Given:

• 
  `\theta` = 35°
• O = 0.795 m
• A = D m

Substitute the values into the formula and solve for D:

`\implies \sf \tan(35^(\circ))=(0.795)/(D)`

`\implies \sf D=(0.795)/(\tan(35^(\circ)))`

`\implies \sf D=1.135377665\:m`

Therefore, the shortest distance that the sofa should be placed from the wall is 1.135 m (3 d.p.).