Steve has 12 biscuits in a tin.
There are 7 digestive and 5 chocolate biscuits.
Steve takes two biscuits at random from the tin.
Work out the probability that he chooses two different types of biscuits.

Answer:

hole at x=-3

Step-by-step explanation:

The hole is the discontinuity that exists after the fraction reduces. (Still doesn't exist for original of course)

The discontinuities for this expression is when the bottom is 0. x^2-9=0 when x=3 or x=-3 since squaring either and then subtracting 9 would lead to 0.

So anyways we have (x+3)/(x^2-9)

= (x+3)/((x-3)(x+3))

Now this equals 1/(x-3) with a hole at x=-3 since the x+3 factor was "cancelled" from the denominator.

Answer:

Step-by-step explanation:

Total number of balls = 3 + 2 = 5

1)

a)

[tex]Probability \ of \ taking \ 2 \ black \ ball \ with \ replacement\\\\ = \frac{3C_1}{5C_1} \times \frac{3C_1}{5C_1} =\frac{3}{5} \times \frac{3}{5} = \frac{9}{25}\\\\[/tex]

b)

[tex]Probability \ of \ one \ black \ and \ one\ white \ with \ replacement \\\\= \frac{3C_1}{5C_1} \times \frac{2C_1}{5C_1} = \frac{3}{5} \times \frac{2}{5} = \frac{6}{25}[/tex]

c)

Probability of at least one black( means BB or BW or WB)

 [tex]=\frac{3}{5} \times \frac{3}{5} + \frac{3}{5} \times \frac{2}{5} + \frac{2}{5} \times \frac{3}{5} \\\\= \frac{9}{25} + \frac{6}{25} + \frac{6}{25}\\\\= \frac{21}{25}[/tex]

d)

Probability of at most one black ( means WW or WB or BW)

[tex]=\frac{2}{5} \times \frac{2}{5} + \frac{3}{5} \times \frac{2}{5} \times \frac{2}{5} + \frac{3}{5}\\\\= \frac{4}{25} + \frac{6}{25} + \frac{6}{25}\\\\=\frac{16}{25}[/tex]

2)

a) Probability both black without replacement

  [tex]=\frac{3}{5} \times \frac{2}{4}\\\\=\frac{6}{20}\\\\=\frac{3}{10}[/tex]

b) Probability  of one black and one white

 [tex]=\frac{3}{5} \times \frac{2}{4}\\\\=\frac{6}{20}\\\\=\frac{3}{10}[/tex]

c) Probability of at least one black ( BB or BW or WB)

 [tex]=\frac{3}{5} \times \frac{2}{4} + \frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{3}{4}\\\\=\frac{6}{20} + \frac{6}{20} + \frac{6}{20} \\\\=\frac{18}{20} \\\\=\frac{9}{10}[/tex]

d) Probability of at most one black ( BW or WW or WB)

 [tex]=\frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{1}{4} + \frac{2}{5} \times \frac{3}{4}\\\\=\frac{6}{20} + \frac{2}{20} + \frac{6}{20} \\\\=\frac{14}{20}\\\\=\frac{7}{10}[/tex]

Answer:

B. 23

Step-by-step explanation:

BC = 32

CA = 44

To find the length of CD, apply the altitude of right triangle formula, (altitude-on-hypotenuse theorem) which is given as:

h = √(xy)

Where,

h = CB = 32

x = CA = 44

y = CD = ?

Plug in the values

32 = √(44 × CD)

Square both sides

32² = 44 × CD

1,024 = 44 × CD

Divide both sides by 44

1,024/44 = CD

CD = 23 units (nearest whole unit)

Answer:

0.001831055

Step-by-step explanation:

Here, n = 16, p = 0.5, (1 - p) = 0.5 and x = 14

As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 14)

P(x =14) = 16C14 * 0.5^14 *(1-0.5)^2

               = 120 * 0.5^14 *(1-0.5)^2

= 0.001831055

Answer:

2

Step-by-step explanation:

The equation of a circle is given as:

    (x-h)^2 + (y-k)^2 = r^2

so r^2 = 4

r = sqrt(4)

r = 2

Answer:

A

Step-by-step explanation:

Answer:

Step-by-step explanation:

M

Answer:

D. y =

Step-by-step explanation:

The solutions to this graph (meaning when y equals 0 or when the graph crosses the x-axis) are 4 and 5.

The only answer choice that has the solutions 4 and 5 when you factor it out is D.

Here's the proof:

[tex]x^{2} -9x + 20[/tex]

Factors of 20: - 5 & -4

Sums that add up to -9: -5 + (-4)

[tex]x^{2} -4x-5x+20[/tex]

(factor the first two terms and the last two terms separately)

[tex](x^{2}-4x)(-5x+20)[/tex]

[tex]x(x-4) -5(x-4)[/tex]

(x - 5) (x - 4)

Hope it helps (●'◡'●)

Answer:

4:30

because 9:30 minus 4:30 = 5:00 and 5:00 minus 30 =4:30

Answer:

[tex]9x - 3 = 78 \\ 9x - 3 + 3 = 78 + 3 \\ 9x = 81 \\ \frac{9x}{9} = \frac{81}{9} \\ x = 9[/tex]

Answer:

[tex]9x - 3 = 78 \\9 x = 78 + 3 \\ 9x = 81 \\ x = \frac{81}{9} \\ x = 9[/tex]

Hello,

I have only found 113 solutions (i have num 15 given)

nb= 1         ::: 27*n +  98= 30*n + 56===> n= 1  4

nb= 2         ::: 28*n +  95= 30*n + 67===> n= 1  4

nb= 3         ::: 29*n +  65= 30*n + 17===> n= 4  8

nb= 4         ::: 29*n +  65= 30*n + 18===> n= 4  7

nb= 5         ::: 29*n +  65= 30*n + 47===> n= 1  8

nb= 6         ::: 29*n +  65= 30*n + 48===> n= 1  7

nb= 7         ::: 29*n +  74= 30*n + 16===> n= 5  8

nb= 8         ::: 29*n +  74= 30*n + 18===> n= 5  6

nb= 9         ::: 29*n +  74= 30*n + 56===> n= 1  8

nb= 10        ::: 29*n +  74= 30*n + 58===> n= 1  6

nb= 11        ::: 30*n +  16= 29*n + 74===> n= 5  8

nb= 12        ::: 30*n +  17= 29*n + 65===> n= 4  8

nb= 13        ::: 30*n +  18= 29*n + 65===> n= 4  7

nb= 14        ::: 30*n +  18= 29*n + 74===> n= 5  6

nb= 15        ::: 30*n +  47= 29*n + 65===> n= 1  8

nb= 16        ::: 30*n +  48= 29*n + 65===> n= 1  7

nb= 17        ::: 30*n +  56= 27*n + 98===> n= 1  4

nb= 18        ::: 30*n +  56= 29*n + 74===> n= 1  8

nb= 19        ::: 30*n +  58= 29*n + 74===> n= 1  6

nb= 20        ::: 30*n +  67= 28*n + 95===> n= 1  4

nb= 21        ::: 36*n +  97= 40*n + 25===> n= 1  8

nb= 22        ::: 38*n +  59= 40*n + 27===> n= 1  6

nb= 23        ::: 38*n +  65= 40*n + 27===> n= 1  9

nb= 24        ::: 38*n +  69= 40*n + 15===> n= 2  7

nb= 25        ::: 39*n +  78= 45*n + 6===> n= 1  2

nb= 26        ::: 39*n +  82= 40*n + 15===> n= 6  7

nb= 27        ::: 39*n +  82= 40*n + 17===> n= 6  5

nb= 28        ::: 39*n +  82= 40*n + 65===> n= 1  7

nb= 29        ::: 39*n +  82= 40*n + 67===> n= 1  5

nb= 30        ::: 40*n +  15= 38*n + 69===> n= 2  7

nb= 31        ::: 40*n +  15= 39*n + 82===> n= 6  7

nb= 32        ::: 40*n +  17= 39*n + 82===> n= 6  5

nb= 33        ::: 40*n +  25= 36*n + 97===> n= 1  8

nb= 34        ::: 40*n +  27= 38*n + 59===> n= 1  6

nb= 35        ::: 40*n +  27= 38*n + 65===> n= 1  9

nb= 36        ::: 40*n +  65= 39*n + 82===> n= 1  7

nb= 37        ::: 40*n +  67= 39*n + 82===> n= 1  5

nb= 38        ::: 46*n +  87= 50*n + 39===> n= 1  2

nb= 39        ::: 46*n +  87= 52*n + 9===> n= 1  3

nb= 40        ::: 47*n +  68= 50*n + 29===> n= 1  3

nb= 41        ::: 47*n +  83= 50*n + 26===> n= 1  9

nb= 42        ::: 47*n +  98= 51*n + 6===> n= 2  3

nb= 43        ::: 47*n +  98= 53*n + 2===> n= 1  6

nb= 44        ::: 48*n +  63= 50*n + 29===> n= 1  7

nb= 45        ::: 48*n +  73= 52*n + 9===> n= 1  6

nb= 46        ::: 49*n +  63= 51*n + 7===> n= 2  8

nb= 47        ::: 49*n +  72= 53*n + 8===> n= 1  6

nb= 48        ::: 49*n +  78= 52*n + 30===> n= 1  6

nb= 49        ::: 49*n +  87= 56*n + 3===> n= 1  2

nb= 50        ::: 50*n +  26= 47*n + 83===> n= 1  9

nb= 51        ::: 50*n +  29= 47*n + 68===> n= 1  3

nb= 52        ::: 50*n +  29= 48*n + 63===> n= 1  7

nb= 53        ::: 50*n +  39= 46*n + 87===> n= 1  2

nb= 54        ::: 52*n +  30= 49*n + 78===> n= 1  6

nb= 55        ::: 57*n +  92= 63*n + 8===> n= 1  4

nb= 56        ::: 58*n +  72= 60*n + 34===> n= 1  9

nb= 57        ::: 58*n +  73= 60*n + 49===> n= 1  2

nb= 58        ::: 58*n +  79= 60*n + 31===> n= 2  4

nb= 59        ::: 58*n +  97= 60*n + 13===> n= 4  2

nb= 60        ::: 59*n +  47= 62*n + 8===> n= 1  3

nb= 61        ::: 59*n +  71= 60*n + 23===> n= 4  8

nb= 62        ::: 59*n +  71= 60*n + 28===> n= 4  3

nb= 63        ::: 59*n +  71= 60*n + 43===> n= 2  8

nb= 64        ::: 59*n +  71= 60*n + 48===> n= 2  3

nb= 65        ::: 59*n +  74= 63*n + 2===> n= 1  8

nb= 66        ::: 59*n +  78= 61*n + 30===> n= 2  4

nb= 67        ::: 59*n +  84= 61*n + 30===> n= 2  7

nb= 68        ::: 59*n +  87= 61*n + 3===> n= 4  2

nb= 69        ::: 60*n +  13= 58*n + 97===> n= 4  2

nb= 70        ::: 60*n +  23= 59*n + 71===> n= 4  8

nb= 71        ::: 60*n +  28= 59*n + 71===> n= 4  3

nb= 72        ::: 60*n +  31= 58*n + 79===> n= 2  4

nb= 73        ::: 60*n +  34= 58*n + 72===> n= 1  9

nb= 74        ::: 60*n +  43= 59*n + 71===> n= 2  8

nb= 75        ::: 60*n +  48= 59*n + 71===> n= 2  3

nb= 76        ::: 60*n +  49= 58*n + 73===> n= 1  2

nb= 77        ::: 61*n +  30= 59*n + 78===> n= 2  4

nb= 78        ::: 61*n +  30= 59*n + 84===> n= 2  7

nb= 79        ::: 65*n +  89= 70*n + 24===> n= 1  3

nb= 80        ::: 68*n +  59= 72*n + 3===> n= 1  4

nb= 81        ::: 68*n +  91= 70*n + 45===> n= 2  3

nb= 82        ::: 69*n +  43= 70*n + 15===> n= 2  8

nb= 83        ::: 69*n +  43= 70*n + 18===> n= 2  5

nb= 84        ::: 69*n +  43= 70*n + 25===> n= 1  8

nb= 85        ::: 69*n +  43= 70*n + 28===> n= 1  5

nb= 86        ::: 69*n +  48= 72*n + 3===> n= 1  5

nb= 87        ::: 69*n +  52= 70*n + 14===> n= 3  8

nb= 88        ::: 69*n +  52= 70*n + 18===> n= 3  4

nb= 89        ::: 69*n +  52= 70*n + 34===> n= 1  8

nb= 90        ::: 69*n +  52= 70*n + 38===> n= 1  4

nb= 91        ::: 69*n +  54= 71*n + 8===> n= 2  3

nb= 92        ::: 69*n +  58= 73*n + 2===> n= 1  4

nb= 93        ::: 69*n +  82= 75*n + 4===> n= 1  3

nb= 94        ::: 69*n +  85= 74*n + 20===> n= 1  3

nb= 95        ::: 70*n +  14= 69*n + 52===> n= 3  8

nb= 96        ::: 70*n +  15= 69*n + 43===> n= 2  8

nb= 97        ::: 70*n +  18= 69*n + 43===> n= 2  5

nb= 98        ::: 70*n +  18= 69*n + 52===> n= 3  4

nb= 99        ::: 70*n +  24= 65*n + 89===> n= 1  3

nb= 100       ::: 70*n +  25= 69*n + 43===> n= 1  8

nb= 101       ::: 70*n +  28= 69*n + 43===> n= 1  5

nb= 102       ::: 70*n +  34= 69*n + 52===> n= 1  8

nb= 103       ::: 70*n +  38= 69*n + 52===> n= 1  4

nb= 104       ::: 70*n +  45= 68*n + 91===> n= 2  3

nb= 105       ::: 74*n +  20= 69*n + 85===> n= 1  3

nb= 106       ::: 76*n +  93= 80*n + 45===> n= 1  2

nb= 107       ::: 79*n +  45= 82*n + 6===> n= 1  3

nb= 108       ::: 79*n +  54= 83*n + 6===> n= 1  2

nb= 109       ::: 80*n +  45= 76*n + 93===> n= 1  2

nb= 110       ::: 87*n +  64= 90*n + 25===> n= 1  3

nb= 111       ::: 87*n +  65= 90*n + 23===> n= 1  4

nb= 112       ::: 90*n +  23= 87*n + 65===> n= 1  4

nb= 113       ::: 90*n +  25= 87*n + 64===> n= 1  3

Answer:

x = 3.7

Step-by-step explanation:

By applying sine ratio for the given angle B,

sin(39°) = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]

sin(39°) = [tex]\frac{AD}{AB}[/tex]

0.6293 = [tex]\frac{AD}{7}[/tex]

AD = 4.41

By applying tangent ratio for the given angle C,

tan(50°) = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]

1.19 = [tex]\frac{AD}{x}[/tex]

1.19 = [tex]\frac{4.41}{x}[/tex]

x = 3.7

Answer:

1, 000 hrs

Step-by-step explanation:

The machine prints,

in 3 hrs = 600 books

in 1 hr = 600/ 3 hrs = 200 hrs.

in 5 hrs = 200 × 5

= 1, 000 hrs

The printing machine will print 1000 books in 5 hours.

Let's calculate how many books the printing machine will print in 5 hours based on the given information.

To do this, we'll use the concept of rates and proportions.

Given that the printing machine can print 600 books in 3 hours, we can set up a rate equation as follows:

Rate of printing = Number of books / Time taken

Let "x" be the number of books the machine will print in 5 hours. We can set up the proportion:

600 books / 3 hours = x books / 5 hours

To solve for "x," we cross-multiply:

3 * x = 600 * 5

Now, let's solve for "x":

3x = 3000

x = 3000 / 3

x = 1000

So, the printing machine will print 1000 books in 5 hours.

Given: Printing machine prints 600 books in 3 hours.

Let the number of books the machine will print in 5 hours be "x."

Using the rate formula, we can set up the proportion:

600 books / 3 hours = x books / 5 hours

Cross-multiplying:

3 * x = 600 * 5

Solving for "x":

3x = 3000

x = 3000 / 3

x = 1000

Hence, the printing machine will print 1000 books in 5 hours.

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Answer:

It would take Al 7.5 days to build the grill alone.

Step-by-step explanation:

Since Tim and Al are bricklayers, and Tim can construct an outdoor grill in 5 days, and if Al helps Tim, they can build it in only 3 days, to determine how long would it take Al to build the grill alone should be done the following calculation:

1/5 + X = 1/3

0.20 + X = 0.333

X = 0.333 - 0.20

X = 0.1333333

X = 1 / 7.5

Therefore it would take Al 7.5 days to build the grill alone.

Answer:

-4

Step-by-step explanation:

2t=-1-7

t=-8/2

t=-4

i am not sure also

Answer:

(a) 1 - (15 C 6) / (30 C 6)

(b)  (5 C 2) x (10 C 2) x (15 C 2) / (30 C 2)

Step-by-step explanation:

Number of  nickels = 5

Number of dimes = 10

Number of quarters = 15

(a) The probability of getting 6 quarters  

= (15 C 6) / (30 C 6)

So, the probability of not getting 6 quarters = 1 - (15 C 6) / (30 C 6)

(b) Probability of getting 2 nickels , 2 dimes and 2 quarters

= (5 C 2) x (10 C 2) x (15 C 2) / (30 C 2)

Answer:

1) C

2) D

3) A

4) B

hope it helps

Answer:

[tex]x=4[/tex]

Step-by-step explanation:

We have the quadratic function:

[tex]\displaystyle y=0.3(x-4)^2-2.5[/tex]

And we want to determine its axis of symmetry.

Notice that this is in vertex form:

[tex]y=a(x-h)^2+k[/tex]

Where (h, k) is the vertex of the parabola.

From our function, we can see that h = 4 and k = -2.5. Hence, our vertex is the point (4, -2.5).

The axis of symmetry is equivalent to the x-coordinate of the vertex.

The x-coordinate of the vertex is 4.

Therefore, the axis of symmetry is x = 4.

Answer:

-2/15

Step-by-step explanation:

14x+x+15=13

Answer:

x = -8

Step-by-step explanation:

based on the picture the top line is equal to 13.

So, 14 + x+ x + 15 = 13

      2x + 29 = 13

      2x = -16

x = -8

11/4

Step-by-step explanation:

to find the minimum value we require to find the vertex and determine if max/min

for a quadratic in standard form ; ax² + bx + c

the coordinate of the vertex is..

xvertex = -b/2a

x² - 3x + 5 is in standard form with a = 1,b = - 3 and c = 5

xvertex = - , -3/2 = 3/2

substitute this value into the equation for y-coordinate

yvertex = ( 3/2 ) ² -3 (3/2) + 5 = 11/4

vertex = ( 3/2, 11/4 )

to determine whether max/min

• if a > 0 then minimum u

• ifa < 0 then maximum n

here a = 1 > 0 hence minimum

minimum value of x² - 3x + 5 is 11/4

hope you understand this :)

Step-by-step explanation:

The ratio is 2 to 5 or 2:5 or 2/5. All these describe the ratio in different forms of fractions. The ratio can consequently be expressed as fractions or as a decimal.

A ratio of 2 : 5 states a comparison between two quantities.

A ratio is a comparison between two similar quantities in simplest form.

Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.

In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.

Given, a ratio 2 : 5.

Suppose it is a ratio of no. of pens to no. of pencils.

So, a ratio 2 : 5 states for every 2 pens there are 5 pencils out of 7 pen and pencils.

We can also write no. of pens = 2/(2+ 5) = 2/7 and for pencils it is 5(2+5)

= 5/7.

Generally, ratios are in simplest form we can have more pens and pencils here but it must be in the multiple of 7.

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Answer:

1

Step-by-step explanation:

Answer:

0.0182 = 1.82% probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

A statistician calculates that 7% of Americans are vegetarians.

This means that [tex]p = 0.07[/tex]

Sample of 403 Americans

This means that [tex]n = 403[/tex]

Mean and standard deviation:

[tex]\mu = p = 0.07[/tex]

[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.07*0.93}{403}} = 0.0127[/tex]

What is the probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%?

Proportion below 7 - 3 = 4% or above 7 + 3 = 10%. Since the normal distribution is symmetric, these probabilities are equal, which means that we find one of them, and multiply by 2.

Probability the proportion is below 4%

p-value of Z when X = 0.04.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{0.04 - 0.07}{0.0127}[/tex]

[tex]Z = -2.36[/tex]

[tex]Z = -2.36[/tex] has a p-value of 0.0091

2*0.0091 = 0.0182

0.0182 = 1.82% probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%.

Answer:

her salary will increase by $ 145 for every week

Step-by-step explanation:

x=1st paycheck (integer).

weekly raise = $ 145.

After completing the 1st week she will get $ (x+145).

Similarly after completing the 2nd week she will get

$ (x + 145) + $ 145.

= $ (x + 145 + 145)

= $ (x + 290)

So in this way end of every week her salary will increase by $ 145.

Answer:

D

y = mx + b

-10 is b because it´s the y intercept (the y value when x is 0).

now, the slope (m) is rise/run:

this is easier graphed, but you can see that the run is 3 (moving sideways on the x axis) and the rise is 2 (going up or down) so its 2/3.

because we are going down on the y axis, the slope is negative (so is the y intercept).

So y = -2/3 -10 is the answer.

[tex]{\boxed{\boxed { ⎆ Answer :- }}} \ [/tex]

[tex]2x + 4 = x + 40 \\ 2x - x = 40 - 4 \\ x = 36[/tex]

ʰᵒᵖᵉ ⁱᵗ ʰᵉˡᵖˢ ツ

꧁❣ ʀᴀɪɴʙᴏᴡˢᵃˡᵗ2²2² ࿐

Answer:

(-2, 2)

Step-by-step explanation:

(-2, 2) is the only ordered pair that makes both inequalities true.

Answer:

B

Step-by-step explanation:

got it right

Answer with Step-by-step explanation:

Complete question:

The coordinates  of the vertices of the triangle are (-8,8),(-8,-4), and. Consider QR the base of the triangle. The measure of the base is b = 18 units, and the measure of the height is h = units. The area of triangle PQR is square units.

Let

P=(-8,8)

Q=(-8,-4)

QR=b=18 units

Height of triangle, h=Length of PQ

Distance formula

[tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Using the formula

Height of triangle, h=[tex]\sqrt{(-8+8)^2+(-4-8)^2}=12units[/tex]

Area of triangle PQR=[tex]\frac{1}{2}\times base\times height[/tex]

Area of triangle PQR=[tex]\frac{1}{2}\times 18\times 12[/tex]

Area of triangle PQR=108 square units

Length of QR=18units

Let the coordinates of R(x,y)

[tex]\sqrt{(x+8)^2+(y+4)^2}=18[/tex]

[tex](x+8)^2+(y+4)^2=324[/tex]

[tex]x^2+64+16x+y^2+8y+16=324[/tex]

[tex]x^2+y^2+16x+8y=324-64-16[/tex]

[tex]x^2+y^2+16x+8y=244[/tex]   ......(1)

Using Pythagoras theorem

[tex]H=\sqrt{P^2+B^2}[/tex]

[tex]H=\sqrt{(18)^2+(12)^2}[/tex]

[tex]H=6\sqrt{13}[/tex]units

[tex](6\sqrt{13})^2=(x+8)^2+(y-8)^2[/tex]

[tex]x^2+64+16x+y^2+64-16y=468[/tex]

[tex]x^2+y^2+16x-16y=468-64-64=340[/tex]

[tex]x^2+y^2+16x-16y=340[/tex] .....(2)

Subtract equation (2) from (1) we get

[tex]24y=-96[/tex]

[tex]y=-96/24=-4[/tex]

Using the value of y in equation (1)

[tex]x^2+16x+16-32=244[/tex]

[tex]x^2+16x=244-16+32[/tex]

[tex]x^2+16x=260[/tex]

[tex]x^2+16x-260=0[/tex]

[tex]x^2+26x-10x-260=0[/tex]

[tex]x(x+26)-10(x+26)=0[/tex]

[tex](x+26)(x-10)=0[/tex]

[tex]x=-26, x=10[/tex]

Hence, the coordinate of R (10,-4) or (-26,-4).

Options:

(2, 2), (3, 1), (4, 2)

(2, 2), (3, –1), (4, 1)

(2, 2), (1, –2), (0, 2)

(2, 2), (1, 2), (2, 0)

Answer:

A. (2, 2), (3, 1), (4, 2)  

Step-by-step explanation:

Given

[tex]y > -\frac{1}{3}x + 2[/tex]

[tex]y < 2x + 3[/tex]

Required

Solve for x and y

To solve this, we make use of graphical method (see attachment for graph)

All points that lie on the shaded region are true for the inequality

Next, we plot each of the given options on the graph

A. (2, 2), (3, 1), (4, 2)

All 3 points lie on the shaded region.

Hence, (a) is true

Answer:

A. (2, 2), (3, 1), (4, 2)  

Step-by-step explanation:

Answer:

C, D, D.

Step-by-step explanation:

Problem 6)

We want to determine the equation of the graphed inequality.

First, let's determine the equation of the line for the inequality. We can see that it passes through the points (-2, 0) and (0, 2). Find the slope:

[tex]\displaystyle m=\frac{\Delta y}{\Delta x}=\frac{2-0}{0-(-2)}=\frac{2}{2}=1[/tex]

So, the slope of the line is one.

And since it passes through the point (0, 2), our y-intercept is two. Therefore, the equation of the line is:

[tex]y=x+2[/tex]

Next, notice that the shaded region is below the line. Also, the line itself is also shaded.

Since the shaded region is below the line, y is less than the graph of the line and since the line itself is shaded, our sign is less than or equal to.

Hence:

[tex]y \leq x + 2[/tex]

Our answer is C.

Problem 7)

We have the inequality:

[tex]-2x+8+5x>2x+1[/tex]

First, solve the inequality. Combine like terms:

[tex]3x+8>2x+1[/tex]

Subtract x from both sides:

[tex]x+8>1[/tex]

And subtract 8 from both sides:

[tex]x>-7[/tex]

Therefore, any value greater than -7 will satisfy the inequality.

Out of the choices, the only choice greater than -7 is -5.

So, our answer is D.

Problem 8)

We have the inequality:

[tex]5x+7\leq 8x-3+2x[/tex]

Again, solve the inequality. Combine like terms:

[tex]5x+7\leq 10x-3[/tex]

Subtract 5x from both sides:

[tex]7\leq 5x-3[/tex]

And add three to both sides:

[tex]10\leq 5x[/tex]

Divide both sides by five:

[tex]2\leq x[/tex]

Flip:

[tex]x\geq 2[/tex]

Therfore, any value greater than or equal to 2 will satisfy the inequality.

Out of the choices, the only choice greater than or equal to 2 is 2.

So, our answer is D.