Derive the equation of the parabola with the focus
is (-7,5) and the directrix of y=-11

In this exercise we have to use the knowledge of vectors to find the dot product between them, in this way we can say that:

[tex]|a|*|b|=42.9[/tex]

Knowing that it was stated by the text that the vectors have the value of :

[tex]a=<4.60,7.20>\\ b=<5.10,2.70> [/tex]

The scalar product, or the inner product, or the dot-product, is by summing the products of the respective directions, will be:

[tex]|a|*|b|=4.6*5.1+7.2*2.7\\ =23.46+19.44\\ =42.9 [/tex]

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Answer:

To make the function a decay function the rate of change in the function A(x) is r<0.              

Step-by-step explanation:

Given : The exponential function [tex]A(x)=P(1+r)^x[/tex]

To find : What value for r will make the function a decay function?

Solution :

The exponential function is [tex]A(x)=P(1+r)^x[/tex],

where,  P is the initial value

and r is the rate of change

The condition of rate of change is

1) The function is an exponential growth if their is positive rate of change i.e r>0.

2) The function is an exponential decay if their is negative rate of change i.e. r<0.

Therefore, To make the function a decay function the rate of change in the function A(x) is r<0.

In this question, you have to solve the equation.

5z – 9 = 36

You transpose the 9 to other side so the equation would look like this:

5z = 36 + 9

5z = 45

Divide 5 in both sides to solve the value of z:

z = 9

The best prediction of the number of gold earrings Prudence will pull out is 5 pairs.

To predict the number of pairs of gold earrings Prudence will randomly pull out of her drawer, we need to find the probability of selecting gold earrings from the total number of options.

Prudence has 20 pairs of gold earrings and 4 pairs of silver earrings, making a total of 24 pairs of earrings in her drawer.

The probability of selecting a gold earring is given by:

Number of gold earrings / Total number of earrings =  20 / 24= 5 / 6.

Therefore, the best prediction of the number of pairs of earrings that will be gold is 5 pairs.

Answer:

Step-by-step explanation:

common ratio of a Geometric series can be found by dividing the second term by first term of the series .

we are given that first term a1= -125 and second term a2 =-25

so the common ratio must be

[tex]r=\frac{a2}{a1}[/tex]

[tex]r=\frac{-25}{-125}[/tex]

[tex]r= \frac{1}{5}[/tex]

Hence the common ratio is 1/5.

The real roots of a function are the rational and the irrational roots of the function

The real roots are: [tex]\mathbf{x = -2}[/tex], [tex]\mathbf{x = \frac 14}[/tex], [tex]\mathbf{x = -3 - 2\sqrt{5}}[/tex] and [tex]\mathbf{x = -3 + 2\sqrt{5}}[/tex]

The equation is given as:

[tex]\mathbf{ 4x^4+31x^3-4x^2-89x+22=0}[/tex]

Factorize

[tex]\mathbf{(x + 2) (4 x - 1) (x^2 + 6 x - 11) = 0}[/tex]

Split

[tex]\mathbf{x + 2 = 0\ or\ 4 x - 1 = 0 \ or\ x^2 + 6 x - 11 = 0}[/tex]

Solve for x

[tex]\mathbf{x = -2\ or\ x = \frac 14 \ or\ x^2 + 6 x - 11 = 0}[/tex]

Solve

[tex]\mathbf{x^2 + 6 x - 11 = 0}[/tex] using the following quadratic formula:

[tex]\mathbf{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}[/tex]

So, we have:

[tex]\mathbf{x = \frac{-6 \pm \sqrt{6^2 - 4 \times 1 \times -11}}{2 \times 1}}[/tex]

[tex]\mathbf{x = \frac{-6 \pm \sqrt{80}}{2}}[/tex]

[tex]\mathbf{x = \frac{-6 \pm 4\sqrt{5}}{2}}[/tex]

[tex]\mathbf{x = -3 \pm 2\sqrt{5}}[/tex]

Hence, the real roots are:

[tex]\mathbf{x = -2}[/tex], [tex]\mathbf{x = \frac 14}[/tex], [tex]\mathbf{x = -3 - 2\sqrt{5}}[/tex] and [tex]\mathbf{x = -3 + 2\sqrt{5}}[/tex]

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Answer:

In seventh time she ran more than one mile from first time she ran.

Step-by-step explanation:

Given Vicki started jogging. The first time she ran she ran 3/16 mile. The second time, she ran 3/8 Mile, and the third time, she ran 9/16 mile.

We have to find when was the first time she ran more than one mile.

The sequence followed is

[tex]\frac{3}{16}, \frac{6}{16}, \frac{9}{16}, \frac{12}{16}, \frac{15}{16}, \frac{18}{16}, \frac{21}{16}....[/tex]

First time she ran more than 1 mile is [tex]\frac{3}{16}+1=\frac{19}{16}[/tex]

In the seventh time she ran [tex]\frac{21}{16}[/tex] miles which is more than one mile from the first time.

Hence, in seventh time she ran more than one mile from first time she ran.

Answer

C.

2.8x + 8.7                  

The height of the building seen by both Jack and Jill is 37m to the nearest whole metre.

Let the distance of Jill from the base of the building be represented with x while the height of the building be y so that by observation we get two right triangles with same opposite sides to the angles formed by Jack and Jill. We can solve for x and y using tangent of the angles as follows:

tan 40 = y/x {opposite/adjacent}

y = xtan40 {cross multiplication}

tan 30 = y/(20 + x)

y = tan30(20 + x)

Thus;

xtan40 = tan30(20 + x)

xtan40 = 20tan30 + xtan30

xtan40 - xtan30 = 20tan30 {collect like terms}

x(tan40 - tan30) = 20tan30

x = 20tan30/(tan40 - tan30)

x = 44.1230m

Putting the 44.1230m for x in y = xtan40 we have;

y = 44.1230m × tan40

y = 37.0236m.

Therefore, the height of the building approximately to the nearest whole metre is 37.

We can use the z-score formula with the known z-score corresponding to a probability of 0.01 to calculate the exhaust NOx emission level. This gives us approximately 0.001 g/mi.

In this scenario, we're dealing with a normal distribution problem wherein we use the formula for z-score to answer it. The z-score formula is =(−)/(/√), where x is the data point, μ is the population mean, n is the size of the data set, and σ is the standard deviation.

In this case, we want to find x, or the NOx emission level, such that the probability of finding an automobile with a level greater than x is 0.01. So, we need to find the z-score that corresponds to a probability of 0.01 and then use it in our formula to solve for x.

Using standard z-score tables or an online z-score calculator, we find that a probability of 0.01 corresponds to a z-score of approximately -2.33 (p and z have opposite signs). We then substitute this z-score, along with the given mean μ=0.03 and standard deviation σ=0.01 into the z-score formula, solving for x. The value of n is 64 (number of vehicles in the fleet).

Inserting these values into the formula gives us = + z/√ =0.03 + (-2.33)*(0.01/√64), which simplifies to =0.03 - (2.33/80) = 0.03 - 0.029125= 0.000875 g/mi. Therefore, the NOx emission level, such that the probability of finding an automobile with a level greater than it, is 0.01 for this fleet is approximately 0.001 g/mi when rounded to three decimal places.

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To find the level l such that the probability that the average NOx level x for the fleet is greater than l is only 0.01, we can use the z-score formula and the properties of the normal distribution. The z-score formula allows us to convert a probability into a corresponding value on the distribution. Rearranging the z-score formula, we can find the value of l.

To find the level l such that the probability that the average NOx level x for the fleet is greater than l is only 0.01, we can use the z-score formula and the properties of the normal distribution. The z-score formula is z = (x - µ) / (σ / √n), where x is the average NOx level, µ is the mean, σ is the standard deviation, and n is the sample size. We want to find the value of l such that the area to the right of l under the normal curve is 0.01.

First, we need to find the z-score corresponding to a right-tail probability of 0.01. This can be done using a z-table or a calculator. In this case, the z-score is approximately 2.326. Now we can use the z-score formula to find the value of l. Rearranging the formula, we have l = µ + (z * σ / √n). Substituting the given values, l = 0.03 + (2.326 * 0.01 / √64). Calculating this, we get l ≈ 0.031.

Therefore, the level l such that the probability that the average NOx level x for the fleet is greater than l is only 0.01 is approximately 0.031 g/mi.

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In the case above, the standard matrix A for T:

T= [0 1]

    [-1 0]

A linear transformation is known to be a kind of a function that exist from one vector point to another and it is one that often respects the linear) structure of all of vector space.

Note that in the linear transformation;

T: R² R²

T=  (x, y), (-x, y)

Since:

(x,y) - (x,-y) - (y,-x)

A= [-1 0]

   [0 1]

A= [-1 0]    =       A= [-x]

   [0 1]                    [y]

Then [tex]T_{b}[/tex] is the reflection of (x- y); Since;

B = [0 1]

    [1 0]

Then [tex]T_{B} (T_{a}(x) )[/tex]  =  [0 1]        =      A= [-x]

                              [0 1]                     [y]

  = [-x]

    [y]

Then: T: = [0 1]                  [x]

                [0 1]                   [y]

Therefore, In the case above, the standard matrix A for T:

T= [0 1]

    [-1 0]

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Answer:

Nothing will be left over.

Step-by-step explanation:

You need to multiply the 175 by 14.25 to get his total income each month.

Then you add all of the expenses of 1250, 200, 350, 400, 225, 300.

Huey earns 2,493.75 every month, and his expenses are 2,725.

2,493.75 - 2725 = a negative rational number

I ain't gonna do it, because he ain't got no money in his wallet.

(I don't talk like that above, I'm just pretending to be an official Texan.)

If he has negative money left over, I don't think he would spend some more tbh, so the answer would be nothing would be left over.

Hope this helps someone in the future!

Answer:

I believe you meant to put " log10 (0.001)= -3

10^-3=0.001

The expression log100.001 = -3 in exponential form is 10^-3 = 0.001. This demonstrates that 10 raised to the power of -3 equals 0.001, showing the inverse relationship between logarithms and exponents.

To express log100.001 = -3 in exponential form, recall that the logarithm is the exponent to which the base, in this case, 10, must be raised to produce the given number. The equation log10(0.001) = -3 tells us that the base 10 raised to the power of -3 equals 0.001.

In exponential form, this can be written as 10-3 = 0.001.The concept of converting a decimal to a negative power of 10 involves moving the decimal point to make the number into a whole number, and the negative exponent indicates how many places the decimal point needs to move to the left. In our case, 10-3 means we move the decimal three places to the left: 1 becomes 0.001.

Final answer:

To make the inequality 43 < y - 30 true, add 30 to both sides to get 73 < y. Any number greater than 73, such as 74, 80, or 90, will satisfy the inequality.

Explanation:

To find values that make the inequality 43 < y - 30 true, you simply need to add 30 to both sides of the inequality. This gives you:

73 < y

This means that any value greater than 73 will make the inequality true. Here are three examples:

You can choose any three numbers greater than 73, and they will satisfy the inequality 43 < y - 30.

Answer:

the answer is  1,437

Step-by-step explanation: