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Nutka1998 [239]
2 years ago
6

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 16 cubic feet per minute. The di

ameter of the base of the cone is approximately three times the altitude. At what rate (in ft/min) is the height of the pile changing when the pile is 2 feet high
Mathematics
1 answer:
kogti [31]2 years ago
6 0

The rate at which the height of the pile is changing when the height is 2 ft. is 0.5659 ft./min. Computed using differentiation.

The sand is falling onto a conical pile, thus we get the pile in the shape of a cone, whose volume is given as:

<u>V = (1/3)πr²h</u>,

where V represents the volume of the cone, r represents its radius, and h represents its height.

In the question, we are given that the rate of piling is 16 ft.³/min, which implies that:

dV/dt = 16 ... (i)

We are also given that the diameter is three times the altitude (height), which can be shown as:

2r = 3h,

or, r  = (3/2)h.

Thus, the volume can now be shown as:

V = (1/3)π{(3/2)h}²h,

or, V = (3πh³)/4.

Differentiating this with respect to time, we get:

dV/dt = (3π/4)(3h²)dh/dt.

Now, we need to calculate the rate of change of height, when the height is  2 feet, thus we take h = 2

Using dV/dt = 16 from (i), we can write:

16 = (3π/4)(3(2)²)dh/dt.,

or, 16 = 9π(dh/dt),

or, dh/dt = 16/9π = 0.5659 ft./min.

Thus, the rate at which the height of the pile is changing when the height is 2 ft. is 0.5659 ft./min.

Learn more about differentiation at

brainly.com/question/13461339

#SPJ4

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A triangle is shown. The table names different transformations that can be applied to the triangle.
Citrus2011 [14]

The given transformations are rigid transformation in which the image

can be located by a formula.

Response:

First reflect across the x-axis then reflect across the y-axis

  • Figure: Second graph

First, rotate 180° counterclockwise around the origin. Then reflect across the x-axis.

  • Figure: Third graph

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  • Figure: First graph

<h3>Which method is used to determine the transformation of an image?</h3>

The coordinates of (x, y) following a reflection across the x-axis is (x, -y)

The coordinates following a reflection across the y-axis is (-x, y)

The points of the triangle following the first transformation is therefore;

(4, 4) → (4, -4) → (-4, -4)

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(6, 6) → (6, -6) → (-6, -6)

The image from the first transformation is (-4, -4), (-8, -4), (-6, -6), which

corresponds to the <u>second graph</u>.

The coordinates following a rotation of (x, y) by 180° about the origin is (-x, -y)

A reflection of the point (-x, -y) about the x-axis is (-x, y).

Therefore;

(4, 4) → (-4, -4) → (-4, 4)

(8, 4) → (-8, -4) → (-8, 4)

(6, 6) → (-6, -6) → (-6, 6)

The image from the second transformation is (-4, 4), (-8, 4), (-6, 6), which corresponds to the <u>third graph</u>.

The coordinates following a translation of 9 units to the left is (x - 9, y)

A reflection of the point (x - 9, y) about the x-axis is (x - 9, -y)

Therefore;

(4, 4) → (4 - 9, -4) = (-5, -4)

(8, 4) → (8 - 9, -4) = (-1, -4)

(6, 6) → (6 - 9, -6) → (-3, -6)

The image from the third transformation is (-13, -9), (-1, -4), (-7, -6), which corresponds to the<u> first graph</u>.

Learn more about rigid transformations here:

brainly.com/question/11408409

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Answer:

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