# MIE 2015/Day 1/Problem 6

## Problem 6

Let $(a,b,c)$ be a geometric progression and $\left(\log\left(\frac{5c}{a}\right),\log\left(\frac{3b}{5c}\right),\log\left(\frac{a}{3b}\right)\right)$ be an arithmetic progression, both in these order, so we can say that $a$, $b$ and $c$:

(a) are the sides of an obtusangle triangle.

(b) are the sides of an acutangle triangle that's not equilateral.

(c) are the sides of an equilateral triangle.

(d) are the sides of a right triangle.

(e) can't be the sides of a triangle.

## Solution

$(a,b,c)\implies\boxed{b^2=ac}$

$\left(\log\frac{5c}{a},\log\frac{3b}{5c},\log\frac{a}{3b}\right)\implies\log\frac{3b}{5c}-\log\frac{5c}{a}=\log\frac{a}{3b}-\log\frac{3b}{5c}$

$\log\frac{3ab}{25c^2}=\log\frac{5ac}{9b^2}$

$\frac{3b}{25c^2}=\frac{5c}{9b^2}$

$3^3b^3=5^3c^3$

$\boxed{3b=5c}$

$3^2b^2=5^2c^2$

$3^2ac=5^2c^2$

$\boxed{9a=25c}$

$9a=25c$

$9a=5\cdot3b$

$\boxed{3a=5b}$

So we got these three relations

$\begin{cases}3b=5c\\9a=25c\\3a=5b\end{cases}$

By these equations we can see that $a > b > c$, so it can't be an equilateral triangle.

First, we must see if $a$, $b$ and $c$ are the sides of a tringle.

$|a-b| < c < a+b$

$\left|a-\frac{3}{5}a\right| < \frac{9}{25}a < a+\frac{3}{5}a$

$\frac{2}{5} < \frac{9}{25} < \frac{7}{5}$

So, $a$, $b$ and $c$ can't be the sides of a triangle. $\boxed{\text{E}}$

Plug Valve Introduction

A plug valve is a quarter-turn hands-on shutoff that makes use of a round or conical connect to allow or stop directly with move with the body. The connect has a directly with complete port opening up. Plug valve can be utilized in on-off, standard modest throttling, and drawing away solutions.

The in person conventional for connect shutoffs is typically related to ANSI Conventional B16.10. Nonetheless, a lot of producers have actually chosen to utilize in person measurements offered entrance shutoffs to make sure that connect shutoffs can be utilized as a substitute for entrance shutoff in current procedure solutions.

Connect Shutoff Make One of the most typical plug valve make enables directly with, 2 method solution, with the closure aspect between of the body. The closure aspect, which is a connect and a sleeve, comes with leading entrance accessibility in the body and is secured by a hood top. A diagrammatic depiction of plug valve is revealed listed below.

Outdoors placement, the flow in the connect lines up with the inlet and electrical outlet ports of the shutoff. When the closure aspect is revolved 90° from the open up placement, the strong section of the connect obstructs the move flow and quits the move of liquid.

For some destructive chemical solutions (such a hydrochloric acid, sulfuric acid etc), plug valve bodies are entirely lined with PTFE, PVDF, PVDC, polyethylene, polypropylene or comparable finishings.

3 method connect shutoffs are likewise widespread, with 3rd port commonly at a best angle from the inlet. With the 3 method make, the closure aspect is utilized to draw away or incorporate the move.

Drawing away Move - Move gets in at the inlet and actions with the connect, which networks the move to among both electrical outlets. When connect is relocated 90°, the move is network to the various other electrical outlet. At the midway placement, move might be similarly split to both electrical outlets. Incorporating Move - Move is routed from 2 inlets to a solitary electrical outlet.

Plug valve manufacturer want to permit the appropriate quarter transform activity without over or under stroking of plug valve, a halt collar plan is utilized. The prevent collar is created to make sure that it fits on top of the connect and therefore transforms with the connect stem. A taken care of essential is offered on the hood top at quarter transform course. As the connect stem is relocated, the set essential maintains the prevent collar and the connect from removaling outdoors the quarter transform vary.

Sorts of Connect Shutoffs Connect shutoffs are offered in numerous kinds as provided listed below:

Lubing Connect Shutoff Non Lubing Connect Shutoff Multi Port Connect Shutoff Eccentric Connect Shutoff Broadening Connect Shutoff These will certainly be reviewed is information is different short posts.

Connect Shutoff Applications Connect shutoffs are frequently utilized in low-pressure-low-temperature solutions, although some high-pressure-high-temperature layouts likewise exists. Connect shutoffs are high ability shutoffs that can be utilized for directional move manage, also in modest vacuum cleaner systems. Connect shutoffs can effectively deal with gas and fluid gas. Connect shutoffs can securely deal with severe temperature level move, such as boiler feed sprinkle, condensate, and various other such aspects. Connect shutoffs can be utilized to manage the move of fluids consisting of put on hold solids, for instance, slurries.

Benefits and Drawbacks of Connect Shutoff

Benefits Connect shutoffs have actually basic make with couple of components. Connect shutoffs open up and shut conveniently (other than big measurements). Repair and maintenance of connect shutoffs can be done at the area of procedure. Connect shutoffs have actually reduced move resistance. Connect shutoffs offer trusted leak-proofness.

Drawbacks Due to high rubbing, initially activity (opening-closing) calls for a big quantity of pressure. NPS 4 (DN100) and bigger connect shutoffs call for a transmission or an actuator. Plug valve shifts are narrowing moves as a result of conic connect. Usually, connect shutoffs expenses are a lot bigger compared to sphere shutoffs.