Mathematics, 29.07.2019 12:00 Star6147

An inverted cone is being drained of water at a rate of 30cm cubed per min. it has a height that is 4 times its radius. how fast is the radius changing when the height of the water is 6cm?

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Assuming the question asks for the rate of change in the radius of the water. $\text{given: } \dfrac{dv}{dt} = -30\text{ cm}^3/\text{min},\ h = 4r \\ \text{unknown: } \dfrac{dr_w}{dt} \text{ when $h = 6$ cm}$ a diagram of the situation has been drawn.similar triangles can be used to relate. there are two different right triangles: the smaller one has sides of 6 cm and $r_{water}$ . the other has sides of $4r$ and $r$ . these two pairs of sides share the same angle, so we can equate the ratio of those sides. $\begin{aligned} \dfrac{6}{r_w} & = \dfrac{4r}{r} \\[6pt] \dfrac{6}{r_w} & = 4 \\ r_w & = \dfrac{6}{4} \\ r_w & = \dfrac{3}{2} \end{aligned}$ the formula for the volume of a cone relates the volume of the water $v$ , the radius of the water $r_w$ , and the height of water $h_w$ . $v = \dfrac{1}{3}\pi r_w^2 h$ since the water follows the shape of the cone, the height of the water has to be 4 times the radius of the water. therefore $h_w = 4r_w$ so $v = \dfrac{1}{3}\pi r_w^2 (4r_w) = \dfrac{4}{3}\pi r_w^3$ differentiate both sides with respect to $t$ (time). $\begin{aligned} \dfrac{d}{dt}v & = \dfrac{d}{dt}\left[\dfrac{4}{3}\pi r_w^3\right] \\ \dfrac{dv}{dt} & =4\pi r_w^2 \dfrac{dr_w}{dt} \\ \dfrac{dr_w}{dt} & = \dfrac{1}{4\pi r_w^2} \cdot \dfrac{dv}{dt}\end{aligned}$ so plugging in the information: $\begin{aligned} \dfrac{dr_w}{dt} & = \dfrac{1}{4\pi \left(\tfrac{3}{2}\right)^2} \cdot (-30) \\ & = \dfrac{1}{9\pi} \cdot (-30) \\ & = -\dfrac{10}{3\pi} \text{ cm/min} \end{aligned}$ the radius is decreasing at a rate of $\dfrac{10}{3\pi} \text{ cm/min}$ , which is approx 1.061 cm/min

Answer from: Quest

answer: the price of one pound cabbage = $ 0.8

the point (10,8) represents that the cost of 10 pounds cabbage is $ 8

step-by-step explanation:

since, here is a proportional relation between the number of pounds of cabbage (x) and the price in dollars (y).

therefore, we can write,

$y\propto x$

⇒ y = k x

where k is the constant of proportionality.

now, point (5,4) is in the line.

thus, it will satisfies the above equation of line.

⇒ 4 = 5 k

⇒ k = 0.8

thus, the equation of the line is,

y = 0.8 x

a) for x = 1,

y = 0.8

therefore, the cost of one pound of cabbage is 0.8 dollars.

b) in the below graph (10,8) represents that the cost of 10 pounds of cabbage is 8 dollars.

Answer from: Quest

$a_1 = 1$ , and

$a_{n+1} = a_n + (n + 1)$ for $n \ge 1$

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote $a_1 = 1$ , $a_2 = 3$ , $a_3 = 6$ , so that we can call

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from $a_1 = 1$ , to get to $a_2 = 3$ , we add $2(=1+1)$