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The profile of a wood trough is an isosceles trapezoid, as the figure

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The profile of a wood trough is an isosceles trapezoid, as the figure  [#permalink]

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New post Updated on: 06 Apr 2019, 00:33
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The profile of a wood trough is an isosceles trapezoid, as the figure above shows. If 7.5 cubic feet of water was poured into
the trough that is flat laid, what is the height of the water level?

(A) 0.25
(B) 0.33
(C) 0.4
(D) 0.5
(E) 0.75

VeritasKarishma can u help?

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Originally posted by MBA20 on 04 Apr 2019, 11:51.
Last edited by MBA20 on 06 Apr 2019, 00:33, edited 1 time in total.
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The profile of a wood trough is an isosceles trapezoid, as the figure  [#permalink]

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New post 05 Apr 2019, 20:52
MBA20 wrote:
The profile of a wood trough is an isosceles trapezoid, as the figure above shows. If 7.5 square feet of water was poured into
the trough that is flat laid, what is the height of the water level?

(A) 0.25
(B) 0.33
(C) 0.4
(D) 0.5
(E) 0.75

VeritasKarishma can u help?




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IMG_0205.jpg
IMG_0205.jpg [ 2.11 MiB | Viewed 493 times ]


The parallel sides of the trapezoid are 1 and 3 and the height is 1. When we drop the altitudes AP and BQ, the trapezoid is split into 3 areas - a square of side 1 and two 45-45-90 triangles. This we can see in the figure above.

Now, say the height of the water is h. The two triangles will still be isosceles since they will be similar to the larger triangles APD and BQC.
If the height is h, the base will be h too since the triangles will be isosceles.

Area of the trapezoid covered by water = Area of rectangle + Area of two 45-45-90 triangles

= h * 1 + 2*(1/2)*h*h = h^2 + h

Volume of the water = Area of trapezoid * 10 = (h^2 + h) *10 = 7.5

10h^2 + 10h - 7.5 = 0
(h + 15/10)*(h - 5/10) = 0
h = 0.5 feet

Answer (D)

Note that volume should be given as 7.5 cubic ft, not square ft

**Edited to add clarity
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Re: The profile of a wood trough is an isosceles trapezoid, as the figure  [#permalink]

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New post 06 Apr 2019, 00:32
Dear VeritasKarishma,

Thanks for your response.

One doubt:- In your diagrammatic presentation, you have assumed that the water level is till the brim. But as shown in the diagram, the shaded region I suppose is the water level.
As per your explanation:-
Area of the trapezoid covered by water = Area of rectangle + Area of two 45-45-90 triangles

= h * 1 + 2*(1/2)*h*h = h^2 + h

So how can height be equal to base?
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Re: The profile of a wood trough is an isosceles trapezoid, as the figure  [#permalink]

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New post 06 Apr 2019, 03:55
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MBA20 wrote:
Dear VeritasKarishma,

Thanks for your response.

One doubt:- In your diagrammatic presentation, you have assumed that the water level is till the brim. But as shown in the diagram, the shaded region I suppose is the water level.
As per your explanation:-
Area of the trapezoid covered by water = Area of rectangle + Area of two 45-45-90 triangles

= h * 1 + 2*(1/2)*h*h = h^2 + h

So how can height be equal to base?


No, the water level is not to the brim. I don't know to what level it is and I have not shown it in the figure.
We know that the height of the trough is 1 foot and we have used that to find that the two triangles APD and BQC will be isosceles right triangles.

So whatever may be the height of the water, the triangles will remain isosceles since they will be similar to the larger triangles APD and BQC.
Since the triangles will be isosceles, if we assume that water is up to height h, the base of the triangle will also be h.
Hence we use (1/2)*h*h
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Re: The profile of a wood trough is an isosceles trapezoid, as the figure  [#permalink]

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New post 06 Apr 2019, 09:55
volume of water in isosceles trapezoid= 0.5*(sum of parallel sides)*height*width
= 0.5*[1+(1+2x)]*x*10
=5x*(2+2x)
volume of water given in question=7.5cu. m
Hence 10x^2 +10x=7.5
or x=0.5m
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Re: The profile of a wood trough is an isosceles trapezoid, as the figure   [#permalink] 06 Apr 2019, 09:55
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