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

Question

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?

KarishmaB · Answer

IMG_0205.jpg 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

MBA20 · Answer

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?

KarishmaB · Answer

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

nick1816 · Answer

volume of water in isosceles trapezoid= 0.5*(sum of parallel sides)*height*width
 = 0.5* *x*10
 =5x*(2+2x)
volume of water given in question=7.5cu. m
Hence 10x^2 +10x=7.5
or x=0.5m

Fdambro294 · Answer

(1st) Guessing/Approximation Method

Volume of a 3-D Prism = (Area of the cross-section) * (Length or extension of that cross-sectional area)

Entire volume of the trapezoid = (area of trapezoid face) * (extension of that face 10 feet in length

= (1/2) (1 + 3) (1) * (10)

= 20

Volume of water = 7.5

7.5 / 20 = .375

IF the front face were symmetric, the water would fill up to top of the trough evenly. The height would be proportional to the ratio of the (volume of water) / (volume of 3-D prism).

In that case, the height would be (.375) of the (1 foot height) =

.375

Since the length of the bottom of the trough is 1 foot, and the cross-section / trapezoid face increases in length as it moves to the top of 3 ft, the water should fill up the trough more quickly in the beginning.

Logically this means that the height will be something greater than > .375 feet

.4 seems too close to an “evenly” divided water and .75 seems too far off.

(D) .5 would be a great guess

(2nd) Algebra

Shown perfectly already by the posters above me.

Drop 2 perpendicular heights on the trapezoid face from the 1 foot base to the 3 foot base.

Creates a square of 1 by 1 in the middle AND two 45-45-90 isosceles right triangles.

The water will fill to a level H feet from the 1 foot base.

Since the water level is Parallel to the 3 foot base, the two triangles created by the water level Line will be similar to the larger isosceles right triangles.

Area of water is thus:

(Area of 2 isosceles right triangles + Area of middle rectangle) * (extension of cross-sectional area of 10 feet)

* (10) = volume of water of 7.5

* (10) = 7.5

(10H) (H + 1) = 7.5

Rather than try to factor the quadratic, plug in the answer choices

(D) .5

(10 * .5) (1 + .5) =

(5) (1.5) = 7.5

*D*

Posted from my mobile device

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

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