## Benefits Of Automatic Bagging Machines

What are Automatic Bagging Machines?
An Automatic Bagging Machine is a mechanism that automates the packaging process in production. The packaging machine automatically inserts the product in a bag or a pouch and seals it. By installing Automatic Bagging Machines in a company’s packaging operations, companies have several advantages on expenses and productivity. Businesses, large or small, are always looking for ways to maximize time and labor while reducing their production cost. Choosing the right type of bag on roll machines that suit business needs can help fulfill these objectives. There will be increased productivity, enhanced consistency, reduction in quality issues, and reduced workstations. Moreover, you will notice the improvement in packaging accuracy and the safety of both products and workers. You can install Automatic Bagging Machines in almost any production facility and use them to pack products such as cosmetics, pharmaceuticals, mechanic parts, food, and beverages.

Benefits of Installing Automatic Bagging Machines
There are numerous benefits associated with the installation of an Automatic Bagging System in production facilities. Here are some of the main advantages.

1. Enhanced Efficiency
Integrating automatic garbage bag machines will help improve the efficiency of operations and your warehouse and reduce potential quality issues.
2. Customizable Automatic Bagging Machines
The automatic cold cutting flat bag machines are customizable mainly to the production line requirement. With automated bagging, you can pack individual parts of a product. Automatic Bagging Machines are also suitable to be used with different packaging materials. Customize the coloring and size of the bags and preprinted styles.
3. Product Safety
In many instances, insufficient packaging may adversely affect the lifespan of goods. This is a common occurrence with products packaged through manual operations. An Automatic Bagging Machine will prevent mistakes. Automation will improve the quality of your packing and eliminate the chances of damage to goods or reduced shelf life.
4. Improved Productivity
Automatic Bagging Machines will reduce the chances of errors in the production line. Compared with manual labour, the productivity level will show improvement due to the speed and consistency associated with automatic bagging systems. Whether you use semi or fully Automated Bagging Machines, there will still be substantially more quantity produced than hand-packaging operations. The Automatic Bagging Machines involve loading a film roll or bagging on the system and pack one product after another quickly. Only when the bagging material runs out will it need someone to refill, saving time and money.

Financial Benefits of Automatic Bagging Machines
Cost savings and coming up with new solutions are always one of the top priorities of businesses. Automatic side sealing bag machines can be a cost-effective investment in the long term. Not only will it improve production volumes but significantly reduce the workforce requirements. Reduce the cost of labour for sorting, processing, folding, and banding the products. For instance, if 20 employees currently working on your production lines handling the packaging and deploying an automatic bagging infrastructure will save you money by reducing the need to have 20 employees working on packaging and increasing production volume translating into increased revenue.
1. Improved Sustainability
One of the best ways is to reduce the need for transportation in your production process. Automatic draw tape garbage bag machines produce uniform packaging that allows more products to fit fewer trucks than hand-made inconsistent bagging. The packaging method helps you ship more items with a lower carbon footprint. Save cost on fuel budget while reducing greenhouse gas emissions.
2. Saving on Material Costs
Most companies have packaging requirements ranging from different bagging sizes. Instead of investing in purchasing and inventory various bags, Automatic Bagging Machines allow you to buy film rolls to make bags of different sizes. Moreover, using thinner gauge packaging film, there can be some additional savings on material cost.

How to Evaluate Best Automatic Bagging Machines for Your Company
Not only is it an effective method, but it also simplifies your packaging operations while saving money on many fronts. However, you must pay due diligence when looking for the right automatic bagging equipment for your company. Speak to the manufacturer about your specific needs and ways to improve the system’s efficiency. Moreover, you will always need employees to operate the machinery. Thus, make sure you pick something easy to use that requires minimal training.

Main features of T-shirt bag making machine:
1. The T-shirt bag making machine is equipped with servo motor to control the bag length. The brand of servo motor is Yaskawa, Japan.
2. A frequency converter is installed on the T-shirt bag making machine to control the speed and reduce the power consumption, making the operation of the T-shirt bag making machine convenient.
3. The one line four lines t-shirt bag machine adopts cold knife cutting design. The cutting blades of T-shirt bag making machine are made of skh3 material.
4. The two lines t-shirt bag making machine is equipped with four sets of sealing strips, two sets with single sealing lines for making bottom sealing bags (opening pockets at the top), and the other two sets with two sealing lines (for sealing bags at the top and bottom) for making semi-finished T-shirt bags. The sealing strip of the T-shirt bag making machine can be easily replaced with shopping bags and ordinary bags.
5. The side gusset t-shirt bag machine is equipped with a photocell (sick) to accurately register the film printing point.
6. The T-shirt bag making machine has a continuous working system and adopts an automatic shutdown alarm device. When the film is empty, the T-shirt bag making machine will stop automatically.

## How Liquid Filling Machines Benefit the Paint and Coatings Industry

How Liquid Filling Machines Benefit the Paint and Coatings Industry
Using Filling Machines to Increase Efficiency and Profitability
Liquid fillers are integral to liquid packaging lines, with automated models capable of maximizing efficiency. Without this equipment, the filling process wouldn’t be reliable enough to ensure that no product loss occurs because of inaccurate fill levels.
Technological developments that are creating more automation and computerizing many of the components in filling machines have made them more dependable than ever before, with many of the best models designed to allow for full customization and user friendliness.
One type of liquid filler that can meet the needs of the paint and coatings industry is the net weigh filling machine. Designed to handle products of low to high viscosity, net weigh fillers are ideal for filling liquids in bulk quantities, such as 5-gallon pails, with consistent weight levels for each container.
Net weigh fillers work by using independently timed valves with custom programming through the filler’s computer. They can then fill precise amounts of liquid by gravity into containers, stopping once the liquid reaches the specified weight.
These fillers can fill many different types and sizes of containers, with many of the top models capable of lasting for many years.
What is a Piston Filling Machine?
At Liquid Packaging Solutions, there are a number of different types of filling machines manufactured to handle different product viscosities, different fill sizes and other variations in packaging projects. The piston filling machine can solve many issues for products with particulates or high viscosity liquids, though it can also handle thin and medium viscosity products as well.
As product sits in the hopper, the valve, which sits between the hopper and the nozzle, will be open from the hopper to the cylinder. The piston will begin to withdraw from the cylinder, typically after an operator activates the fill by stepping on a foot switch. As the piston withdraws, product from the hopper will fill the empty cylinder. Once the piston has withdrawn to the desired point, the valve will rotate to allow product to move through the nozzle. At this point the piston push back in to the cylinder and move product through the nozzle and in to the waiting bottles or other containers. This process creates a highly accurate volumetric fill as the interior volume of the cylinder will never change, meaning the volume of product released to the bottles will never change.
The hopper sizes can vary from project to project based on the size of the containers or fills. Not all piston filling machines, and in particular, the automatic piston fillers, will use a hopper from which to pull product. Automatic lines will likely include a tank or pull from a bulk source. The cylinder and piston combination are also available in different sizes to accommodate different projects. The speed with which the piston moves can be adjusted, different piston sizes can be used to meet volume requirements and even multi-piston, automatic filling machines can be designed for use with inline packaging systems. LPS piston fillers allow the operator to adjust the length of the piston stroke, which in turn adjusts the volume of product that is pulled in to the cylider with each fill cycle. This way a single piston size can handle a range of container sizes. While multiple strokes of the piston can also be used for larger fills, at some point the efficiency of using multiple strokes will become low enough that simply changing out the piston for large containers will be the better solution.
The nozzle used on any piston filler will be chosen to meet the needs of the particular project at hand. For instance, a product with large chunks of fruit or vegetables will not work well if a narrow nozzle is used to move product in to the bottles. On the other hand, a very large nozzle will be cumbersome with a small mouthed bottle. There is virtually no limit to the type of nozzles that can be used, including custom manufactured nozzles where special projects are concerned.
Though a simple concept, the piston filler can be an ideal solution for many projects and for liquids thick and thin. Though these machines are known for handling viscous products, in the right circumstances they will handle free-flowing liquids as well. For assistance finding the best type of filling machine for your own packaging project, contact Liquid Packaging Solutions today.
Overflow fillers, gravity fillers, automatic piston filling machine and other liquid fillers all vary in the way that they move product into a bottle or container. However, the automatic versions of these machines almost always have certain features in common. These features are intended to add efficiency, consistency and reliability to the packaging equipment. Below are a few of the most common features of found on Liquid Packaging Solutions’ bottle fillers.
Heavy Duty and Portable Stainless Steel Frame
For consistent and reliable fills, the machine must be stabile throughout the process. The heavy duty stainless steel frame protects against shifting, vibrating and other movement that might effect the volume or the level of the fill, while also avoiding splashes and spills. The stainless steel material is compatible with a vast majority of products, though there are exceptions. When corrosive liquids are run on the machinery, other construction materials may be used for the frame, including HDPE. Ultimately, the material used will be that material which will better extend the useful life of the equipment.
Many packagers fill more than a single product, or at the very least fill into bottles of multiple sizes and shapes. Changing over from one product or bottle to another means stopping production on the liquid filler. These machines include simple adjustments to minimize downtime and maximize production. Fill heads can typically be moved using simple fingertip adjustment knobs, while power height comes standard on automatic equipment, allowing up and down movement with the flip of a switch. Even auxillary equipment such as power conveyors include knob adjustments or other simple components for railing and other changes. Other adjustments, such as time and delay settings, can easily be made from the operator control panel, discussed in more detail below.

## Automatic Cutting and Stripping Machines

Improved technology enables fast cutting, clean stripping and simple blade changeover for various size wires.
Without a sculptor, a piece of clay or marble can never reach its full artistic potential. Rotary, V and die blades in automatic cutting and stripping machines serve a similar role to help conductive wire and cable achieve its full electric potential as part of a harness.
Within one or two seconds, these blades precisely cut each wire or cable to a predetermined length and remove its insulation to expose one or more inner conductors. The wires or cables are then manually or automatically crimped by terminal crimping machine before being brought to the assembly workstation, where assemblers use boards to carefully build each harness.
At Gruber Communications, based in Phoenix, workers assemble lots of cable harnesses for use in data centers every day. The company’s priority since day one has been to produce high-quality cables—and make sure that no cable conductor, or high voltage cable machine is ever nicked or blemished during wire cutting and stripping machine’s processing.
For more than a decade, Gruber workers used separate pneumatic machines to cut and strip each cable. Eventually, though, CEO Pete Gruber grew tired of the constant maintenance on the machines’ check valves and cylinders. This led him to purchase the all-electric EcoStrip 9300 cut and strip machine in 1998.
Made by Schleuniger AG of Switzerland, the machine’s reliability and infrequent need for parts has enabled Gruber to substantially increase its cable harness production over the past 18 years. In fact, this machine continues to precisely cut and strip cables after more than 6 million runs.
Being able to run reliably for nearly 20 years and cut and strip millions of cables or wires is quite common for today’s automatic machines. There are two reasons for this, say suppliers. First is stateof- the art blade technology, which enables fast cutting, clean stripping and simple blade changeover for various size wires. Equally important are operators who understand, implement and optimize each machine’s cutting and stripping capabilities.
FROM SIMPLE TO PROGRAMMABLE
More than 90 years ago, Haaken Olsen—an up-andcoming engineer at Artos Engineering Co.—noticed an increased usage of insulated copper wire in automobiles, appliances and radios. He also saw assembly workers manually measuring wire to predetermined lengths, cutting it and removing the insulation from both wire ends.
Believing manufacturers would be interested in buying an automated machine that could perform this work faster, better and more cost-effectively, Olsen went about developing one. In 1926, Artos introduced the CS-1, the first-ever automatic CAS machine. Olsen vowed to sell at least a dozen, but things went much better than planned. A new industry was born, and Artos alone has sold nearly 100,000 wire processing machines over the past nine decades.
“Cutting and wire stripping machine machines from the 1920s to the 1950s featured mechanical designs,” explains John Olsen II, president of Artos since 2005 and great-grandson of Haaken. “Typically, three pair of fixed-position blades were used to cut and strip the wire. All setup changes were done mechanically by adjusting cams and moving blade spacers.”
More-advanced electropneumatic CAS machines appeared in the 1960s and 1970s, allowing for push-button control of feeding lengths. Since then, according to Olsen, CAS machines have evolved in three areas to become much more efficient.
One is the improved operator interface, which increases the machine’s capability to process small batch sizes and provides full integration with a marking system (laser, inkjet, hotstamp) or slitting device. Another is the use of servomotors for all wire movements to increase processing precision and speed. The third is faster machine changeover by using quick-change guides and blades, and technology like the Artos Sencor system to automate wire setup.
Semi- and fully automatic CAS machines come in three sizes: benchtop, midsize and large. A benchtop model is best for low-volume and prototyping applications. It usually requires little setup, plugs into a standard 110- volt outlet, and is simple to operate (push buttons, small display, limited programming).
Despite being an entry-level machine, the benchtop EcoStrip 9380 from Schleuniger can process single wires from 30 to 8 AWG and two wires (up to 0.12-inch diameter) in parallel. It is operated via S. ON software on a 5.7- inch color touch screen, and features the company’s Bricks electronic platform for precise wire feeding by using automatic wire prefeeder. An optional belt feeding system can be set for normal, roller or short mode processing.
Midsize machines are designed for medium-volume applications, which suppliers define as processing up to a few thousand wires or cables per week. These machines may or may not be standalone, but they are bigger and offer more programming options than benchtop models.
One such unit is the CS-326 from Artos. The fully electric, servo-driven machine processes wire and cable from 30 to 4 AWG or 0.5 inch OD. It cuts wire to a length of 0.25 inch to 3,250 feet. Minimum and maximum stripping lengths are 0.01 inch and 39 inches, respectively.
The machine features the Sencor system that senses the conductor within the wire and automatically sets blades at the proper stripping diameter. This technology reduces wire waste, shortens setup time and monitors blade wear.
Separate accessories enable the unit to cut Kevlar-insulated wire and strip coaxial and ignition cables. An optional work table lets companies easily move the 400-pound machine to any workstation.
Schleuniger offers six versions of its MultiStrip 9480 machine to cover a wide range of applications (32 to 8 AWG wire) and budgets. Four models (MR, RS, RSX and RX) feature a fully programmable rotary incision unit capable of processing coaxial and multilayer cables. A multiposition indexing cutter head, standard on all models except the S, accepts blade cassettes that change out quickly and easily. The machine cuts and strips wire as short as 2.3 inches and as long as 3,281 feet. In short mode, wires as short as 0.375 inch, with a 0.125-inch strip length on each end, can be processed.
Large machines are for high-volume (up to several thousand pieces per shift) processing of singleconductor wire as large as 4/0 AWG, and multiconductor or shielded cable up to 1.5 inches OD. These standalone units feature large cutter heads, infeed and outfeed mechanisms, an HMI and multiple protocol interfaces. Users of these machines usually require one to two days of hands-on operations training by the supplier.
Most large machines can also be networked with other assembly machines via a plant’s ERP and MES software. Manufacturers especially like this capability because it provides full traceability for every job, and enables them to track how many cycles each machine has completed and when maintenance should be scheduled.
Artos’ CS-327 machine processes cables as large as 4/0 AWG or 1.37 inches in diameter, including battery and welding cables, power cables for appliances, and multiconductor cables for signal and power. The unit’s dualblade cutter head and belt infeed and outfeed systems are servo-driven. Minimum wire cut length is 10 inches in standard mode and less than 2 inches in short mode. Strip lengths are programmable to 40 inches.
Also standard are an integrated length encoder for accuracy and quality, an HMI for PC operator control and a removable wire scrap collection tray. Options include a three-blade cutter head for high-speed processing and special tooling for steel cables.
FEWER CHALLENGES THAN BEFORE
“In the 1950s, the average harness in an American car contained fewer than 50 wires,” notes Rob Boyd, senior product manager at Schleuniger. “Today’s car features many harnesses that have hundreds of wires of varying gauges and lengths. As a result, harness makers need versatile automatic cutting machine and stripping machines to meet this challenge.”
They also need to make sure that their machine operators are trained to understand the dynamics that exist between wire insulation (depending on wire supplier), nonsymmetrical wire, and blade design and performance limitations. Tim Crider, sales director at Komax Wire, cites as an example the lower margin of error when processing PVC-insulated wire as compared to Teflon-insulated wire. Because PVC is softer and less challenging to cut and strip, the operator doesn’t need to pay as close attention to process parameters, blade positioning and wear.
Komax’s Kappa 331 machine addresses these and many other challenging applications. It processes wire from 24 to 2 AWG and cable up to 0.63 inch OD in large and small batches. The unit also performs full and partial pull-off operations on single conductors and individual coax layers, and strips the outer jackets from cables with or without shielding.
A key feature is the Kappa Sensorik laser sensor, which automatically detects the wire conductor and uses inductive measuring to determine its diameter. The sensor and a chargecoupled device (CCD) line optically measure the outside cable diameter and then check that the cable is present during processing. This feature greatly shortens setup time and changeover, and reduces operating errors.
For the past 18 months, a large wire harness and cable manufacturer has been using the Kappa 322 machine to cut and strip three-conductor 14 AWG cable (40 inches long) at a rate of 600 pieces per hour (pph). This midsize unit processes wire from 30 to 4 AWG and enables easy setup and changeover without tools.
“Buying a midsize machine to constantly perform heavy-duty work is a common problem,” says Armando Zacarias, sales and service manager at Eubanks Engineering Co. “A machine that’s capable of processing 32 to 8 gauge wire is really not designed to process 8 gauge wire all day long. Using the machine that way will likely require it to often be refurbished or rebuilt. A better approach is to buy a machine that’s able to process wire as large as 4 gauge.”
Operators use a cassette to quickly insert and remove blades from Eubanks’ fully programmable AirStrip 7400 machine. Microprocessor-controlled and easy to operate, the machine handles stranded conductor wire from 32 to 8 AWG, and multiconductor cable up to 0.31 inch OD. It strips cable up to 20 inches long, and can be programmed to do step and center stripping.
A bit larger in size is the more powerful 2700-05. It cuts and strips wire from 32 to 8 AWG, as well as multiconductor and flat cable up to 0.31 inch wide. Operators input wire processing parameters on the built-in keypad. Zacarias says consumer electronics and automotive manufacturers use this machine in high-volume, low-mix environments because of its high production rate (up to 10,800 pph).
Another ongoing challenge is making the wire and cable as straight as possible before it enters the CAS machine. Suppliers often provide material on the smallest spool possible, which, unfortunately, results in bent wire and cable that may require a straightener.
To avoid this extra processing step, Boyd recommends thin wire be wrapped around spools at least 10 inches in diameter. Thicker wire and cable should be delivered on much wider barrels, so that it unwinds in a large loop that is easy to straighten.

## Undecidability of the language of all Turing Machines with repeat strings as their language

Show that the language consisting of all Turing machines whose language consists of strings that can be broken up into two consecutive and equal strings is undecidable.

I would prefer if reduction was used and not Rice’s theorem.

## Transforming multi-tape Turing machines to equivalent single-tape Turing machines

We know that multi-tape Turing machines have the same computational power as single-tape ones. So every $$k$$-tape Turing machine has an equivalent single-tape Turing machine.

About the computability and complexity analysis of such a transformation:

Is there a computable function that receives as input an arbitrary multi-tape Turing machine and returns an equivalent single-tape Turing machine in polynomial time and polynomial space?

## Is LXD virtualization as isolated and secure as KVM virtual machines?

Stumbled upon a privacy conscious hosting provider that uses LXD vs KVM to manage user VPS instances.

My understanding is KVM is more isolated so using LXD doesn’t make sense from a privacy perspective.

Perhaps I am missing something?

## How to prove the language of all Turing Machines that accept an undecidable language is undecidable?

I want to prove that $$L=\{\langle M \rangle |L(M)\text{ is undecidable}\}$$ is undecidable

Suppose L is decidable. Let $$E$$ be the decider from $$L$$. Let $$A$$ be a TM which is recognizing $$A_{TM}$$. Let $$S$$ be a TM which works on input $$\langle M,w \rangle$$ in the following way:

1. Construct a TM $$N$$ which works on Input $$x$$ as follows: Run $$M$$ on $$w$$. If $$M$$ $$accepts$$ run $$A$$ on $$x$$ and accept $$x$$ if $$A$$ accepts.(In this case is $$L(N)=A_{TM}$$). If $$M$$ $$rejects$$ $$w$$, $$accept$$ $$x$$.(In this case is $$L(N)=\Sigma^*$$)
2. Run $$E$$ on $$N$$ and accept if N accepts. Otherwise reject

I am not sure if my reduction is the right way or not. Maybe someone can help to finish the reduction 🙂

## Relations between deciding languages and computing functions in advice machines

I’m trying to understand implications of translating between functions and languages for P/Poly complexity. I’m not sure whether the following all makes sense. Giving it my best shot given my current understanding of the concepts. (I have a project in which I want to discuss Hava Siegelmann’s analog recurrent neural nets, which recognize languages in P/Poly, but I’d like to understand and be able to explain to others implications this has for computing functions.)

Suppose I want to use an advice Turing $$T_1$$ machine to calculate a function from binary strings to binary strings $$f: {0,1}* \rightarrow {0,1}*$$. $$T_1$$ will be a machine that can compute $$f$$ in polynomial time given advice that is polynomial-size on the length of arguments $$s$$ to $$f$$, i.e. $$f$$ is in P/Poly. (Can I say this? I have seen P/Poly defined only for languages, but not for functions with arbitrary (natural number) values.)

Next suppose I want to treat $$f$$ as defining a language $$L(f)$$, by encoding its arguments and corresponding values into strings, where $$L(f) = \{\langle s,f(s)\rangle\}$$ and $$\langle\cdot,\cdot\rangle$$ encodes $$s$$ and $$f(s)$$ into a single string.

For an advice machine $$T_2$$ that decides this language, the inputs are of length $$n = |\langle s,f(s)\rangle|$$, so the relevant advice for such an input will be the advice for $$n$$.

Question 1: If $$T_1$$ can return the result $$f(s)$$ in polynomial time, must there be a machine $$T_2$$ that decides $$\{\langle s,f(s)\rangle\}$$ in polynomial time? I think the answer is yes. $$T_2$$ can extract $$s$$ from $$\{\langle s,f(s)\rangle\}$$, and then use $$T_1$$ to calculate $$f(s)$$, and then encode $$s$$ with $$f(s)$$ and compare it with the original encoded string. Is that correct?

Question 2 (my real question): If we are given a machine $$T_2$$ that can decide $$\{\langle s,f(s)\rangle\}$$ in polynomial time, must there be a way to embed $$T_2$$ in a machine $$T_3$$ so that $$T_3$$ can return $$f(s)$$ in polynomial time?

I suppose that if $$T_2$$ must include $$T_1$$, then the answer is of course yes. $$T_3$$ just uses the capabilities of $$T_1$$ embedded in $$T_2$$ to calculate $$f(s)$$. But what if $$T_2$$ decides $$L(f)$$ some other way? Is that possible?

If we are given $$s$$, we know its length, but not the length of $$f(s)$$. So in order to use $$T_2$$ to find $$f(s)$$, it seems there must be a sequential search through all strings $$s_f = \{\langle s,r\rangle\}$$ for arbitrary $$r$$. (I’m assuming that $$f(s)$$ is unbounded, but that $$f$$ has a value for every $$s$$. So the search can take an arbitrary length of time, but $$f(s)$$ will ultimately be found.)

One thought I have is that the search for a string $$s_f$$ that encodes $$s$$ with $$f(s)$$ has time complexity that depends on the length of the result $$f(s)$$ (plus $$|s|$$, but that would be swamped when $$f(s)$$ is long).

So now the time complexity does not have to do with the length of the input, but only the length of $$f(s)$$. Maybe $$L(f)$$ is in P/Poly if $$f$$ is in P? (Still confused here.)

Thinking about these questions in terms of Boolean circuits has not helped.

## Intuition for Church-Turing thesis for Turing machines

I can very clearly see "why" mu-recursion is a universal model of computation, i.e. why the Church-Turing thesis — that any physically computable algorithm can be executed with mu-recursion — holds for mu-recursion. It reflects exactly the type of algorithms that I can carry out with my own brain.

I cannot see an analogous intuition for understanding why the Turing machine can execute any physically computable algorithm — i.e. how did Turing "see" that the Turing machine was a good definition to use? Is there a good way to "imagine" the algorithms I perform in terms of the Turing machine, as opposed to general recursion as I am used to?

## Scheduling jobs online on 3 identical machines – a lower bound of 5/3

Consider the Online Scheduling Problem with $$3$$ identical machines. Jobs, with arbitrary size, arrive online one after another and need to be scheduled on one of the $$3$$ machines without ever moving them again.

How can I show, that there can’t be any deterministic Online-Algorithm which achieves a competitive ratio of $$c<\frac{5}{3}$$.

This should be solved by just giving some instance $$\sigma$$ and arguing that no det. algorithm can do better. Same can easily be done for $$2$$ machines and $$c<\frac{3}{2}$$. Sadly I can’t find any solution to this (more or less) textbook answer.