Which of the following relations is a function?

Ответ:

ZachLaVine2016

03.08.2022

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

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Which of the following relations is a function? OA. (2.0), (-1, 3), (7, 1), (-1,5)
OB. (2,4), (-1, 2), (7, 1), (-8, 2)
OC. (7,1),(-1, 4), (2, 1), (7, 2)
OD. (2, 4), (-1, 6), (2, 3), (-8, 2)

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Which of the following relations is a function? OA. (2.0), (-1, 3), (7, 1), (-1,5) OB. (2,4), (-1, 2), (7, 1), (-8, 2) OC. (7,1),(-1, 4), (2, 1), (7, 2) OD. (2, 4), (-1, 6), (2, 3), (-8, 2)

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Which of the following relations is a function? OA. (2.0), (-1, 3), (7, 1), (-1,5)
OB. (2,4), (-1, 2), (7, 1), (-8, 2)
OC. (7,1),(-1, 4), (2, 1), (7, 2)
OD. (2, 4), (-1, 6), (2, 3), (-8, 2)