Find the LU factorization of L -4 A= 20 4 بر تن د That is, write A = LU where L is a lower triangular matrix with ones on the diagonal, and U is an upper triangular matrix. A=

Answers

Answer 1

The LU factorization of L -4 A= 20 4 بر تن د is:
A = LU, where L =   1    0    0
                  4    1    0

and U =   1    0    0
         0    1    0
         0    0    20.

To find the LU factorization of L -4 A= 20 4 بر تن د, we can use Gaussian elimination.

First, we can subtract 4 times the first row from the second row to eliminate the -4 in the (2,1) position:
L -4 A=    1    0    0       20
         -4    1    0       4 بر تن د

becomes:
L -4 A=    1    0    0       20
          0    1    0       84 بر تن د

Now, we can see that L will be:
L =   1    0    0
     4    1    0

To get U, we can continue with Gaussian elimination. The pivot element in the (2,2) position is 1, so we don't need to do anything there. However, we need to eliminate the 4 in the (2,1) position. To do this, we can subtract 4 times the first row from the second row:

L -4 A=    1    0    0       20
          0    1    0       84 بر تن د
becomes:
L -4 A=    1    0    0       20
          0    1    0       4 بر تن د

Now, we can see that U will be:
U =   1    0    0
     0    1    0
     0    0    20

Therefore, the LU factorization of L -4 A= 20 4 بر تن د is:
A = LU, where L =   1    0    0
                  4    1    0

and U =   1    0    0
         0    1    0
         0    0    20.

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Related Questions

A normal population has mean µ = 51 and standard deviation σ = 19. Find the value that has 25% of the population above it. Round the answer to at least one decimal place.
The value that has 25% of the population above it is_____

Answers

The value that has 25% of the population above it is approximately 64.1.

To find the value that has 25% of the population above it, we can use the Z-score formula and the standard normal distribution.

The Z-score formula is given by:

Z = (X - µ) / σ

Where:

Z is the Z-score,

X is the value we want to find,

µ is the population mean, and

σ is the population standard deviation.

To find the value with 25% of the population above it, we need to find the Z-score corresponding to the 75th percentile. The 75th percentile corresponds to a cumulative probability of 0.75.

Using a Z-table or a Z-score calculator, we can find the Z-score that corresponds to a cumulative probability of 0.75, which is approximately 0.6745.

Now, we can rearrange the Z-score formula to solve for X:

Z = (X - µ) / σ

Rearranging, we have:

X = Z * σ + µ

Substituting the values we have:

X = 0.6745 * 19 + 51

X ≈ 13.1295 + 51

X ≈ 64.13

Rounded to at least one decimal place, the value that has 25% of the population above it is approximately 64.1.

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factor. (help plz!)

x^2+7x+10

Answers

Answer:

\((x + 5) \times (x + 2)\)

Step-by-step explanation:

\( {x}^{2} + 7x + 10\)

\( {x}^{2} + 5x + 2x + 10\)

\(x \times (x + 5) + 2(x + 5)\)

\((x + 5) \times (x + 2)\)

ANSWER:

Hope this helps you

factor. (help plz!)x^2+7x+10

we need the (x,y) to tell us where?

Answers

Answer:

a point on a two-dimensional graph

Step-by-step explanation:

x is up and down, y is left and right

X is the horizontal shift of the point, and y is the vertical shift of the point.

malia bought a block of wax to make candles. she used 3.5 pounds of the wax to make vanilla-scented candles and has less than 5 pounds of wax left to make orange-scented candles. let x represent the weight of the block of wax originally. which inequality describes the problem?

Answers

The required inequality describes for the given problem is  x < 8.5.

What is inequality?

The idea of inequality, which is the state of not being equal, especially in terms of status, rights, and opportunities, is at the core of social justice theories. However, because it frequently has diverse meanings to different people, it is prone to misunderstanding in public discourse.

According o question:

she used 3.5 pounds of the wax to make vanilla-scented candles and has less than 5 pounds of wax left to make orange-scented candles.

let x represent the weight of the block of wax originally

then,inequality can be written as,

x - 3.5 < 5

x < 8.5

Thus, required inequality is x < 8.5.

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what is the answer i need them now pls I'm going to fail

what is the answer i need them now pls I'm going to fail

Answers

Answer:

A

Step-by-step explanation:

You said fast so no explanation

Its A!! I hope you dont fail

plz help me :) it asking for what q=

plz help me :) it asking for what q=

Answers

Answer:

\(q = \frac{4A}{h} - b\)

Step-by-step explanation:

Please see the attached picture for full solution.

plz help me :) it asking for what q=

Casey is conducting an organizational analysis for BMH Corp. She finds that the company's long-term goal of being ranked as number one in pet food sales is not being reached. Casey's organizational analysis should also consider:

Answers

In addition to assessing the company's progress towards its long-term goal of being ranked as number one in pet food sales, Casey's organizational analysis should consider other factors that may be affecting the company's performance.

Casey's organizational analysis should include a comprehensive evaluation of various aspects of BMH Corp. to identify potential factors contributing to the company's inability to achieve its long-term goal. This analysis may involve examining internal factors such as the company's structure, culture, leadership, and decision-making processes. It may also involve assessing external factors such as market conditions, competition, customer preferences, and industry trends.

By considering these additional factors, Casey can gain a holistic understanding of the company's strengths, weaknesses, opportunities, and threats. This analysis will enable her to identify areas that require improvement or changes in strategy to help BMH Corp. move closer to its long-term goal. Additionally, it may reveal insights into potential barriers or challenges that need to be addressed in order to achieve sustainable growth and competitive advantage in the pet food industry.

By conducting a comprehensive organizational analysis, Casey can provide valuable insights and recommendations to BMH Corp. for enhancing its performance and aligning its efforts with the objective of becoming the leading player in pet food sales.

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suppose that two boys named davis, three boys named jones, and four boys named smith are seated at random in a row containing nine seats. what is the probability that the davis boys will occupy the first two seats in the row, the jones boys will occupy the next three seats, and the smith boys will occupy the last four seats?

Answers

The probability that the Davis boys will occupy the first two seats in the row, the Jones boys will occupy the next three seats, and the Smith boys will occupy the last four seats is 7.937×\(10^{-4}\).

There are \(\left[\begin{array}{ccc}9\\2,3,4\\\end{array}\right]\) potential arrangements for the nine boys if we do not make a distinction amongst boys with the same last name. We are curious in the likelihood of a specific one of these configurations.

Probability = \(\frac{1}{\left[\begin{array}{ccc}9\\2,3,4\\\end{array}\right]}\)

                  = \(\frac{2!3!4!}{9!}\)

                 = 0.0007937

                = 7.937×\(10^{-4}\)

Hence, the probability is 7.937×\(10^{-4}\).

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the screamers are coached by coach yells at. the screamers have $12$ players, but two of them, bob and yogi, refuse to play together. how many starting lineups (of $5$ players) can coach yells lot make, if the starting lineup can't contain both bob and yogi? (the order of the $5$ players in the lineup does not matter; that is, two lineups are the same if they consist of the same $5$ players.)

Answers

The total number of starting lineups (of 5 players) can coach yells lot make, if the starting lineup can't contain both bob and yogi is 672.

What is starting lineups?

An official list of the players who will take part in the event when the game starts is known as a starting lineup in sports. The players who are in the starting lineup are frequently referred to as starters, while the rest are known as bench players or replacements.

Case 1: Bob starts (and Yogi doesn't). In this case, the coach must choose 4 more players from the 10 remaining players (remember that Yogi won't play, so there are only 10 players left to select from). Thus there are 10C4 lineups that the coach can choose.

⇒ ¹⁰C₄ = 10!/4!(10-4)! = 10!/4!6!

⇒ ¹⁰C₄ = (10*8*9*7)/(4*3*2*1)

⇒ ¹⁰C₄ = 210

Case 2: Yogi starts (and Bob doesn't). As in Case 1, the coach must choose 4 more players from the 10 remaining players. So there are 10C4 lineups in this case.

⇒ ¹⁰C₄ = 210

Case 3: Neither Bob nor Yogi starts. In this case, the coach must choose all 5 players in the lineup from the 10 remaining players. Hence there are 10C5 lineups in this case.

⇒ ¹⁰C₅ = 10!/5!(10-5)! = 10!/5!5!

⇒ ¹⁰C₅ = (10*8*9*7*6)/(5*4*3*2*1)

⇒ ¹⁰C₅ = 252

To get the total number of starting lineups, we add the number of lineups in each of the cases:

Case 1 + Case 2 + Case 3

= 210 + 210 + 252 = 672

The total number of starting lineups (of 5 players) can coach yells lot make, if the starting lineup can't contain both bob and yogi is 672.

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what is the answer to this
12b−15>21

Answers

Answer:

b > 3

Step-by-step explanation:

\(12b-15>21\\\\12b-15+15>21+15\\\\12b>36\\\\\frac{12b>36}{12}\\\\\boxed{b>3}\)

Hope this helps.

What is the distance between the points (4, 6) and (5, 3)? Write the answer in simplest radical form.

Answers

Answer:

\( \sqrt{10} \: \: units\)

Step-by-step explanation:

\( \sqrt{(6 - 3)^{2 } + (4 - 5)^{2} } \)

\( \sqrt{9 + 1} \)

\( \sqrt{10} \)

= 3.16 units

The distance between two points of a line is given as:

Distance =  \(\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}\)

Where,

\((x_1, y_1)~~and ~~(x_2, y_2)\) are the two points.

The simplest radical form of the distance between (4, 6) and (5, 3) is√10.

What is the distance between two points of a line?

The distance between two points of a line is given as:

Distance =  \(\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}\)

Where,

\((x_1, y_1)~~and ~~(x_2, y_2)\) are the two points.

We have,

The two points.

(4, 6) and (5, 3)

Now,

Distance = \(\sqrt{(5-4)^2 +(3-6)^2}\)

Distance = \(\sqrt{1^2 +(-3)^2}\)

Distance = \(\sqrt{1 + 9}\)

Distance = √10

We can also write this as,

Distance = \(10^{1/2}\)

Thus,

The simplest radical form of the distance between (4, 6) and (5, 3) is√10.

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on a number line what is the distance between -29and 100

Answers

Answer:

The total distance is 129

Step-by-step explanation:

In order to answer this question you will need to find 2 distances and then add them to each other.

Begin at -29.  How far will you have to move to the right 29 in order to get to 0?

Now how far will you have to more to get from 0 to 100?

Draw an open number line to help you.

_____________________________________________________

-29                                           0                                                   100

Your first distance is 29.  Your second distance is 100.

The total distance is 129.

A ball is thrown vertically upward with an initial velocity of 64 feet per second. The distance s​ (in feet) of the ball from the ground after t seconds is s=64t-16t^2 (a) At what time t will the ball strike the​ ground? ​(b) For what time t is the ball more than 48 feet above the​ ground?

Answers

Given:

The distance s​ (in feet) of the ball from the ground after t seconds is

\(s=64t-16t^2\)

To find:

(a) At what time t will the ball strike the​ ground?

(b) For what time t is the ball more than 48 feet above the​ ground?

Solution:

(a)

We have,

\(s=64t-16t^2\)

Substitute s=0, to find the time t when the ball strike the​ ground.

\(0=64t-16t^2\)

\(0=16t(4-t)\)

Using zero product property, we get

\(16t=0\Rightarrow t=0\)

\(4-t=0\Rightarrow t=4\)

Therefore, the ball strike the​ ground in initial condition (t = 0) and after 4 seconds (t = 4).

(b)

Now, s > 48, to find the time t when the ball will be more than 48 feet above the​ ground.

\(64t-16t^2\geq 48\)

\(0> 16t^2-64t+48\)

Divide both sides by 16.

\(0> t^2-4t+3\)

\(0> t^2-t-3t+3\)

\(0> t(t-1)-3(t-1)\)

\(0>(t-1)(t-3)\)

Related equation is \((t-1)(t-3)=0\). Zeroes are t=1,3. These two number divide the number line is three parts. (-∞,1),(1,3),(3,∞)

\(0>(t-1)(t-3)\) inequality is true for only (1,3).

It is only possible when t lies in the interval (1,3).

Therefore, the ball will be more than 48 feet above the​ ground between 1 and 3 seconds.

Evaluate f(1/2) if f(x)=-3x^3 + 2x^2. Enter fractions using “/“

f(1/2)=

Evaluate f(1/2) if f(x)=-3x^3 + 2x^2. Enter fractions using /f(1/2)=

Answers

exact form: 1/8
decimal form: 0.125

what is ten more than 90

Answers

What is ten more than 90?

90+10=100

Answer:

100

Step-by-step explanation:

90 + 10

Find the present value of 30 annual payments of $4,500 per annum where the first payment is made 5 years from now. So there are 30 annual payments from t=5 to t=34 inclusive. The discount rate is 9% pa. The present value of these payments is: Select one: a. $49,999.998 b. $47,550.695 c. $35,421.2587 d. $33,686.1107 e. $32,751.5184

Answers

The present value of 30 annual payments of $4,500 per annum where the first payment is made 5 years from now with a discount rate of 9% per annum is $49,999.998. Therefore, the correct option is a. $49,999.998.

Present Value (PV) is the value at which the money that is expected to be received at some future time is worth in today's dollars. It is also called the "discounted value." It is used to measure the value of a future amount in today's dollars. Therefore, it is an essential component of discounted cash flow analysis, used to calculate the value of an investment or debt instrument over time.

The formula to calculate the present value of annuity formula is as follows: PV = [PMT * (1 - (1 / (1 + r)n))]/r

Where,PV = present value of annuity

PMT = payment made every year

r = rate of interest per year

n = number of years

An annuity is a fixed sum of money paid each year.

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if a rigid motion was used to transform Image A into Image B and then a rigid motion was used to transform Image B into Image C, would Imagine C have to be the same size and shape as Image A? Explain.​

if a rigid motion was used to transform Image A into Image B and then a rigid motion was used to transform

Answers

Answer:

Yes

Step-by-step explanation:

A rigid motion is a transformation that preserves the size and shape of the original image. A rigid motion was used to transform image A into image B and then a rigid motion was used to transform image B into image C.

The only change experienced by objects which undergo rigid motion or transformation is the relative change in the position of the objects such that the side which was upward before now face downward and other similar changes. Hence, objects which undergo rigid motion do not experience any change in size or shape.

Rigid transformation may occur in the form of rotation, reflection or translation.

During rigid motion, the original shape and size of the object is preserved as the angles and side lengths are unaltered.

Therefore, since the object A underwent rigid motion to form B, then B also underwent rigid motion to form C, we can conclude that the size and shape of image A will be preserved in image C.

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if a rigid motion was used to transform Image A into Image B and then a rigid motion was used to transform
if a rigid motion was used to transform Image A into Image B and then a rigid motion was used to transform

8x−2=−11+7x +2
Please help me!!

Answers

Answer:

x=-7

Step-by-step explanation:

8x - 2 = -11 + 7x + 2

First, separate the terms: move x to one side

subtract 7x from both sides

8x -7x - 2 = -11 + 7x -7x + 2

x - 2 = -11 + 2

Now get x by itself by adding 2 to both sides

x - 2 +2 = -11 + 2 +2

x = -11 + 4

simplify

x = -7

you're welcome

124{n} = 232{5} find n​

Answers

Answer:

n=9.354

Step-by-step explanation:

n=

\( \frac{290}{31} \)

n=9×

\( \frac{11}{31} \)

So n=9.354

15) Suppose that Tony ordered 3 pizzas. Each pizza has 8 slices. Being shared with 5 people (3 pizzas - 8 slices in each
pie... total: 24 PIECES OF PIZZA)

Answers

Answer:

4.8 slices each

Step-by-step explanation:

8*3= 24

24/5= 4.8

Select EVERY indeterminate form to which L'Hopital's Rule can be directly applied to (there may be more than one). 0.00 0° U U 1° oo U U 8 - 0 U 8|8 o 8

Answers

The indeterminant form to which the L'Hopital's Rule can be applied are 0/0 and ∞/∞ , the correct options are (d) and (f) .

L'Hopital's Rule in calculus is used to evaluate weird limits that result in indeterminate forms that are 0/0, ∞/∞ .  These types of limits

(a) cannot be calculated by direct substitution of limit and/or

(b) can be evaluated but using a very long procedure .

that means if \(\lim_{x \to a} \frac{f(x)}{g(x)}\) = 0/0  or ∞/∞ ,

then \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\)

In the question ,

there are several indeterminant forms given which are  0.∞ ,  \(0^{0}\) , \(1^{\infty}\) , 0/0 ,  ∞ - ∞ , ∞/∞ ,  ∞⁰  .

By the definition of L'Hopital's Rule , the option that matches with the definition is 0/0 and ∞/∞ ,

So , the correct option is (d) 0/0 and (f) ∞/∞   .

Therefore , The indeterminant form to which the L'Hopital's Rule can be applied are 0/0 and ∞/∞ , the correct options are (d) and (f) .

The given question is incomplete , the complete question is

Select EVERY indeterminate form to which L'Hopital's Rule can be directly applied to (there may be more than one).

(a) 0.∞

(b) \(0^{0}\)

(c) \(1^{\infty}\)

(d) 0/0

(e) ∞ - ∞

(f) ∞/∞

(g) ∞⁰

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an urn contains 6 blue balls and 4 red balls. 6 balls are chosen at random and without replacement from the urn. if x is the number of red balls chosen, find the standard deviation of x.

Answers

The standard deviation of the number of red balls chosen, x, is 1.2.

In probability theory, the standard deviation is a measure of the amount of variation or dispersion in a set of values. It helps us understand how spread out the values are from the mean. In this scenario, we have an urn containing blue and red balls, and we are randomly selecting 6 balls without replacement. We want to determine the standard deviation of the number of red balls chosen, denoted by x.

To find the standard deviation of x, we need to calculate the variance first. The variance represents the average of the squared differences between each value and the mean. The standard deviation is then the square root of the variance.

Step 1: Finding the probability of selecting a red ball

In the given urn, there are a total of 6 blue balls and 4 red balls. To determine the probability of selecting a red ball, we divide the number of red balls by the total number of balls:

P(red) = 4 / (6 + 4)

          = 0.4

Step 2: Calculating the mean of x

The mean of x, denoted as μx, represents the expected value or average number of red balls chosen. Since the probability of selecting a red ball is 0.4, we can calculate the mean as follows:

μₓ = (Number of trials) × P(red)

   = 6 × 0.4

   = 2.4

Step 3: Determining the variance of x

The variance, denoted as Var(x), is calculated by finding the average of the squared differences between each value of x and the mean:

Var(x) = (Number of trials) × P(red) × P(blue)

          = 6 × 0.4 × 0.6

          = 1.44

Step 4: Finding the standard deviation of x

Finally, the standard deviation, denoted as σx, is the square root of the variance:

σₓ = √Var(x)

    = √1.44

    = 1.2

Therefore, the standard deviation of the number of red balls chosen, x, is 1.2.

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There are 60 picture books in the library. The picture books make up 15% of all the books in the library. What is the total number of books in the library?

Answers

Answer: 400

Step-by-step explanation: 60 divided by 15% = 400 and i also took the test

There are 60 picture books in the library. The picture books make up 15% of all the books in the library.

The total number of books in the library is 400 books.

Given that there are 60 picture books in the library, and these said 60 books happen to make up 15% of the collection in the entire library. To find the total number of books in the library, we then say that

Total number of books in the library = total number of picture books in the library ÷ percentage of picture books

Total number of books in the library = 60 / 0.15

Total number of books in the library = 400.

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the world record for the most number of speed skips is 60 seconds is 332 skips if the record holder skipped at a constant ratio of seconds to skips how many makes in 15 seconds

Answers

The record holder skips 83 times in 15 seconds

We refer to a ratio as constant if the number we multiply by remains constant throughout the series.

By comparing two amounts of the same units, the ratio can be used to calculate how much of one item is included in the other. There are two different categories for ratios. The first is a part-to-part ratio, whereas the second is a part-to-whole ratio.

The World record for the most number of skips is 332 in 69 seconds.

Let's assume the record holder skipped at a constant ratio.

The proportion, if s is the number of skips:

skips / seconds = s / 15 = 332/60

s / 15 = 332 / 60

s = 332( 15/60 )

s = 332/4

s = 83

She skipped 83 times in the space of 15 seconds.

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The price of a sweater was reduced from $20 to $12 by what percentage was the price of the sweater reduced

Answers

It should be a 40% reduction :)

If price  of a sweater was reduced from $20 to $12 then the percentage was the price of the sweater reduced is 40%.

What is Percentage?

percentage, a relative value indicating hundredth parts of any quantity.

Given that the price of a sweater was reduced from $20 to $12.

Original price is 20.

new price is 12.

We have to find the how much  percentage was the price of the sweater reduced.

To do this we have to find the difference between prices over the original price and multiply with 100.

Percentage = 20-12/20×100

=8/20×100

=40%

Hence, if price  of a sweater was reduced from $20 to $12 then the percentage was the price of the sweater reduced is 40%.

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(x-3)(x+4) please send the help

Answers

Answer:

x^2 + x -12

Step-by-step explanation:

I am going to expand and simplify because you haven't mentioned what to do:

(x-3)(x+4)

x^2 +4x -3x- 12

x^2 +x-12

Need ASAP
Select the correct answer.
Which expression is equivalent to this quotient?

Need ASAP Select the correct answer.Which expression is equivalent to this quotient?

Answers

Your answer is : D.)

Hope this helps love! <3

The expression that is equivalent to the given quotient of the algebraic expression is; C: 3(x - 1)

How to simplify algebraic expressions?

We are given the algebraic expression;

[(3x² - 3)/(x² + 3x)]/[(x + 1)/(x(x + 3))]

The algebra can be simplified as;

[3(x² - 1)/x(x + 3)] * (x(x + 3))/(x + 1)

x(x + 3) will cancel out to give;

3(x² - 1) * 1/(x + 1)

This can be simplified further to get;

3(x + 1)(x - 1) * 1/(x + 1)

⇒ 3(x - 1)

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Given the least-squares regression line, , what is the predicted amount of an unstable element, in grams, that is left after 8 years?

Answers

Taking into account the least-squares regression line ln(element)=2.304-0.101(time). 1.496 grams of an unstable element will still be present after 8 years.

Given that,

Taking into account the least-squares regression line ln(element)=2.304-0.101(time).

We have to find how many grams of an unstable element will still be present after 8 years.

The equation is ln(elements)=2.304-0.101(time) with the element and time.

ln is nothing but a natural logarithm.

So,

We substitute 8 years in place of time.

ln(element)=2.304-0.101(time).

ln(element)=2.304-0.101(8)

ln(element)=2.304-0.808

ln(element)=1.496

Therefore, 1.496 grams of an unstable element will still be present after 8 years.

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find a vector equation for the line segment from (4, −3, 5) to (6, 4, 4). (use the parameter t.)

Answers

Thus,  the vector equation for the line segment is: r(t) = (4, -3, 5) + t(2, 7, -1), 0 ≤ t ≤ 1

To find the vector equation for the line segment from (4, -3, 5) to (6, 4, 4), we need to first find the direction vector and the position vector.

The direction vector is the difference between the two points:
(6, 4, 4) - (4, -3, 5) = (2, 7, -1)

Next, we need to choose a point on the line to use as the position vector. We can use either of the two given points, but let's use (4, -3, 5) for this example.

So the position vector is:
(4, -3, 5)

Putting it all together, the vector equation for the line segment is:
r(t) = (4, -3, 5) + t(2, 7, -1), 0 ≤ t ≤ 1

This equation gives us all the points on the line segment between the two given points. When t = 0, we get the starting point (4, -3, 5), and when t = 1, we get the ending point (6, 4, 4).

Any value of t between 0 and 1 gives us a point somewhere on the line segment between the two points.

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Lamont has purchased 20 trading cards and wants to have at least 50 trading cards. Write and solve an inequality to nd the number of trading cards Lamont needs. Select all of the true statements.

Answers

Given:

Lamont has purchased 20 trading cards.

He wants to have at least 50 trading cards.

To find:

The inequality for the number of trading cards Lamont needs and solve it.

Solution:

Let x be the number of trading cards Lamont needs.

He has 20 trading cards. So,

Total cards = x + 20

It is given that, he wants to have at least 50 trading cards. It means, total card must be greater than or equal to 50.

\(x+20\geq 50\)

Subtract 20 from both sides.

\(x+20-20\geq 50-20\)

\(x\geq 30\)

Therefore, the required inequality is \(x+20\geq 50\) and solution is \(x\geq 30\).

The number of trading cards that Lamont needs will be at least 30 trading cards.

From the information given, we are informed that Lamont has purchased 20 trading cards and wants to have at least 50 trading cards.

Therefore, the number that will be needed more will be:

= 50 - 20 = 30

Therefore, he'll need at least 30 more cards.

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