A father is four times as old as his son. In 4 years, the father will be three times as old as the son. How old is each now?

Answer:

a. 0.25

Step-by-step explanation:

The given expression is

[tex]\lim_{x \to 2} \frac{x-2}{x^2-4}[/tex]

Factor the denominator using difference of two squares.

[tex]\lim_{x \to 2} \frac{x-2}{(x-2)(x+2)}[/tex]

Cancel out common factors;

[tex]\lim_{x \to 2} \frac{1}{(x+2)}[/tex]

We plug in 2 to get

[tex]\lim_{x \to 2} \frac{1}{(x+2)}=\frac{1}{2+2}[/tex]

[tex]\lim_{x \to 2} \frac{1}{(x+2)}=\frac{1}{4}=0.25[/tex]

Answer:

The correct answer option is A. 0.25.

Step-by-step explanation:

We are given the following expression and we are to find its limit:

[tex] lim_\left \{ {{ x = 2 } [/tex] [tex] \frac { x - 2 } { x ^ 2 - 4 } [/tex].

First of all, we will simplify the expression by factorizing the term in the denominator:

[tex] \frac { x - 2 } { x ^ 2 - 4 } = \frac { x - 2 } { ( x - 2 ) ( x + 2 ) }[/tex]

Cancelling the common terms to get:

[tex]\frac{1}{x+2}[/tex]

Substituting the given value for x to get:

[tex]\frac{1}{2+2}[/tex] = 0.25

The answer is C because it’s length x width x height

Answer:

C) V = (10)(5)(2)

Step-by-step explanation:

V = (10)(5)(2) To find the volume of a right rectangular prisms, multiply the length times the width times the height. -- V = lwh.

The answer is: True.

A counterexample is a way that we can prove that something is not true about a mathematical expression or equation, it could be considered as an exception to a rule, so to check if the answer is true or false, we must prove that a counterexample of the given expression is 0°, so:

The given expression is:

[tex]sinytany=cosy[/tex]

Then,

[tex]sin(0)*\frac{sin(0)}{cos(0)}=cos(0)\\\\0*\frac{0}{1}=1\\\\0*0=1\\0=1[/tex]

We can see that the equation is not fulfilled, so 0° is a counterexample of [tex]sinytany=cosy[/tex] and the answer is true.

Have a nice day!

Answer:

x = 1.723

Step-by-step explanation:

The  zeros of a function f(x) are the points where the function crosses the x-axis. At these points, the function will have a value of zero, that is;

f(x) = 0

We simply graph the function and determine the points where it crosses the x-axis. From the attachment, f(x) crosses the x-axis at;

x = 1.723

Answer:

Step-by-step explanation:

We have given the function:

f(x)=-3x³+6x+5.

We have to find the zeros of the function.

The zeros of the function are the points where the function cuts the x-axis.

And at that points function has value zero.

So to find these points put f(x)= 0 we get,

-3x³+6x+5=0

As from the graph the function cuts the x-axis at x = 1.723 so, this is the zero of the function.

The answer is 2x−5

see the image

chur answer is gonna be 63787 km

Answer:

The expression would be 43x + 128

Step-by-step explanation:

43 dollars is multiplied by however many feet is used.  Then, the exception(the gate) is added to the charge that is 43x

[tex] \cos t (\sec t - \cos t) = \sin^2 t [/tex]

[tex] \cos t (\dfrac{1}{\cos t} - \cos t) = \sin^2 t [/tex]

[tex] \dfrac{\cos t}{\cos t} - \cos^2 t = \sin^2 t [/tex]

[tex] 1 - \cos^2 t = \sin^2 t [/tex]

Use the identity: [tex] \sin^2 t + \cos^2 t = 1 [/tex] and solve for [tex] \sin^2 t [/tex]. You get: [tex] \sin^2 t = 1 - \cos^2 t [/tex]

Do the substitution on the left side to get:

[tex] \sin^2 t = \sin ^2 t [/tex]

[tex] \cos \: t ( \sec \: t - \cos \: t) = \sin^{2}t \\ \\ 1. \: 1 - \frac{ \cos \: t }{ \sec \: t } = \sin^{2} t \\ 2. \: 1 - \cos \: t \cos \: t = \sin^{2} t \\ 3. \: 1 - \cos^{2} t = \sin^{2} t \\ 4. \: 1 - \cos^{2} t - \sin^{2} t = 0 \\ 5. \: 1 - 1 = 0 \\ 6. \: 0 = 0 \\ 7. \: infinitely \: many \: solutions[/tex]

Answer:

9.48

Step-by-step explanation:

The perimeter of a rectangle is the distance around the the shape. It can be found by adding each side of the shape to find a total. In contrast, the area of a shape is the amount inside a shape. It is found using multiplication or A = l*w. Since the width is w and the length is twice its width or 2w, use these expressions in the area formula to find their amount. Then add them together to find the perimeter.

[tex]A = l*w5 = w(2w)\\5 = 2w^2\\2.5 = w^2\\1.58 = w[/tex]

The width of the rectangle is w = 1.58. The length is 2w = 2 (1.58) = 3.16.

Add these amounts by P = l + l + w + w to find the perimeter.

P = 3.16 + 3.16 + 1.58 + 1.58 = 9.48

To find a coterminal angle to 800 degrees within the 0 to 360-degree range, divide 800 by 360 and find the remainder, which is 80 degrees. Hence, 80 degrees is the angle that's coterminal with 800 degrees.

To find an angle that is coterminal with 800 degrees, which means an angle that shares the same terminal side as the 800-degree angle but is between 0 degrees and 360 degrees, we use the concept of coterminal angles. Since a complete circle has 360 degrees, we can subtract or add multiples of 360 degrees to any angle to find a coterminal angle.

Here’s how we find it for 800 degrees:

Therefore, 80 degrees is the angle between 0 degrees and 360 degrees that is coterminal with an angle measuring 800 degrees.

Fine the average rate of change by the difference in N over the difference in T:

3000 - 500 / 4-0 = 2500 / 4 = 625

This means for every 1 T, N = 625.

Now multiply 625 by 8:

625 x 8 = 5000

When T = 8 N = 5,000

The decay rate can be found by dividing -ln(2) by the half life.

Decay rate = -ln(2) / 5715 = -0.00012

Answer:

18.

Step-by-step explanation:

Mean of the values = (70+95+35+90+85) / 5

=  75.

Mean absolute deviation  :-

70 - 75 = |-5|  = 5  (absolute value is taken

95 - 75 = 20

35-75 = 40

90-75 = 15

85-75 = 10

Mean Absolute Deviation = (5 + 20 + 40 + 15 + 10) /  5

= 18.

Answer: 18

Step-by-step explanation: it was correct on ttm

Answer:

The total number of digits required in numbering the pages of a book that has 53 pages would be: 97.

Step-by-step explanation:

This answer would be 97 due to the first nine digits being singular.

The next 44 digits are double, so therefore, you multiply 44 by 2, and you end up with 88.

After that, you take 88 and add the 9, which, is 97 digits in total that you would need to number a 53 page book.

I think the answer is x > -8.

Answer:

x > - 8

Step-by-step explanation:

Given

- 2x + 7 < 23 ( subtract 7 from both sides )

- 2x < 16 ( divide both sides by - 2 )

Remembering to reverse the inequality symbol when dividing/ multipkying by a negative quantity, hence

x > - 8 ← reversed symbol

Answer:

Option D. [tex]42.4\ in^{3}[/tex]

Step-by-step explanation:

we know that

The volume of the cone (juice treats) is equal to

[tex]V=\frac{1}{3}\pi r^{2}h[/tex]

we have

[tex]r=3/2=1.5\ in[/tex] ----> the radius is half the diameter

[tex]h=3\ in[/tex]

assume

[tex]\pi=3.14[/tex]

substitute

[tex]V=\frac{1}{3}(3.14)(1.5)^{2}(3)=7.065\ in^{3}[/tex]

Multiply the volume of one juice treats by 6

[tex](6)7.065=42.39=42.4\ in^{3}[/tex]

Answer:

True.

Step-by-step explanation:

Volume of a cylinder = π r^2 h   where h = height and π r^2 = the base area.

Volume / height =  π r^2 h  /  h  =   π r^2 = base area.

The base area of a cylinder is not the quotient of its volume and its height; it is determined by the area of its circular base. Understanding the relationship between base area and volume in a cylinder is essential in mathematics.

False. The base area of a cylinder is not the quotient of its volume and height. The base area of a cylinder is the same as the area of its circular base, which is equal to πr2 where r is the radius of the base.

The volume of a cylinder is calculated by multiplying the base area with the height of the cylinder. Thus, base area ≠ volume / height.

When the height of a cylinder is equal to its diameter, it minimizes the surface area, but it does not affect the calculation of the base area itself.

Answer:

Step-by-step explanation:

The formula for the surface area of a sphere is ...

  A = 4πr²

For a radius of 6 meters, that is ...

  A = 4π(6 m)² = 144π m²

___

The formula for the volume of a sphere is ...

  V = (4/3)πr³ = A·(r/3)

For the same sphere, this is ...

  V = (144π m²)(6 m/3) = 288π m³

Answer:

Step-by-step explanation:

Is a ≤ b < c are the side lengths of a right triangle, then

[tex]a^2+b^2=c^2[/tex]

A. 6cm, 8cm, 10cm.

Substitute:

[tex]6^2+8^2=10^2\\36+64=100\\100=100\to TRUE[/tex]

B. 12cm, 35cm, 37cm.

Substitute:

[tex]12^2+35^2=37^2\\144+1225=1369\\1369=1369\to TRUE[/tex]

C. 4cm, 6cm, 10cm.

It's not the side lengths of a triangle, because 4 + 6 = 10.

Without looking at it, we will check equality

[tex]4^2+6^2=10^2\\16+36=100\\52=100\to FALSE[/tex]

D. 10cm, 24cm, 26cm.

Substitute:

[tex]10^2+24^2=26^2\\100+576=676\\676=676\to TRUE[/tex]

The measurement that could not represent the side length of a right triangle is  4 cm, 6 cm, 10 cm.

A right triangle is a triangle that consists of three sides. The longest side is the hypotenuse. The other sides are the base and the length. The sum of angles in a triangle is 180 degrees.

According to this theorem, the square of the hypotenuse should be equal to the sum of the square of the other two sides.

The Pythagoras theorem: a² + b² = c²

where:

a = length

b = base

c = hypotenuse

6² + 8² = 10²

36 + 64 = 100

Option B = 12² + 35

= √144 + 1225

= √1369

= 37

Option C = 4² + 6²

16 + 36

= √52

= 7.2

Option D: 10² + 24² = 26²

100 + 576

= √676

= 26

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Answer:

Academy A, Arithmetic, and difference makes most sense to me.

Academy A's professional tennis player numbers can be described by an arithmetic sequence due to a constant increase of 10 every year. In contrast, Academy B's numbers can be represented by a geometric sequence because the increase each year is proportionate to the current number of students, at a rate of a 10% increase per year.

The number of players who went on to play professionally at Academy A can be represented by an arithmetic sequence because the numbers of players who went on to play professionally in successive years have a common difference of 10. For Academy A, each year the number increased by a consistent amount, 10 more players, hence the use of an arithmetic sequence.

On the other hand, the number of players who went on to play professionally at Academy B can be represented by a geometric sequence because the numbers of players who went on to play professionally in successive years have a common ratio of 10%. For Academy B, the rate of increase is proportional to the current number of students, which makes it a geometric sequence.

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Final answer:

The explicit formula for the salary increases represented as a geometric sequence is a_n= 38,000(1.035)^(n-1), and the recursive formula is a_1= $38,000 and a_n= a_(n-1) × 1.035 for n > 1, reflecting an annual raise of 3.5%.

Explanation:

The starting salary at a company is $38,000 per year with an annual raise of 3.5%. To represent the salary increases as a geometric sequence, we need both explicit and recursive formulas.

The explicit formula for a geometric sequence is an= a1 × r(n-1), where a1 is the first term, r is the common ratio, and n is the term number. In this case, a1= $38,000 and r = 1.035 (since the salary increase is 3.5%, or 0.035 in decimal form, plus the original amount, hence 1 + 0.035). Therefore, the explicit formula is an= 38,000(1.035)(n-1).

The recursive formula defines each term based on the previous term. It can be written as a1= $38,000, an= a(n-1) × 1.035, for n > 1. This means that starting from the second year (when n > 1), the salary is obtained by multiplying the previous year's salary by 1.035.

Answer:

x = -1

Step-by-step explanation:

-2 to 2 is 4 units, so -3x + 1 = 4.

We subtract 1 from both sides to get -3x = 3.

Then we divide each side by -3 to get x = -1.

After 10 hours the temperature is shown on the graph as 30 degrees.

50 degrees - 30 degrees = 20 degrees.

The temperature dropped 20 degrees in 10 hours.

Divide the change by the time:

20 degrees / 10 hours = 2 degrees per hour.

Because the temperature dropped, the change would be negative.

The answer is D. -2 degrees per hour.

Answer:

The answers for your two questions are the following

a)11.606 m

b) As the number of swings approaches infinity

Step-by-step explanation:

We are dealing with an exponential equation  series, where the length of  each swing can be represented as

l = [ 15 *(0.95)^(n-1) ]

n is the corresponding number of the swing.

So, for the first swing, n = 1

[ 15 *(0.95)^(1-1) ] = 15 m

a) How far with the pendulum travel on its 6th swing?

We just need to evaluate the previous formula for n = 6

[ 15 *(0.95)^(6-1) ] =

[ 15 *(0.95)^(5) ] =

[ 15 *(0.7737) ] =

[ 15 *(0.7737) ] = 11.606 m

b) How far will the pendulum swing before it essentially stops? Hint: This is an infinite geometric series.

We previously stated that the length of the arc of each swing can be represented as

l(n)  = [ 15 *(0.95)^(n-1) ]       ,      for n>=1

Since the function approaches zero if and only if n approaches infinity, we can say that the pendulum never stops.

Of course, this only happens mathematically, we can always fin a threshold for which the movement cannot be registered anymore.

Please see attached graph for a representation of the function

Answer:

a) It will travel approx 79.47 meters,

b) It will travel 300 meters.

Step-by-step explanation:

Given,

The initial distance travel on first swing = 15 meters,

Also, Each subsequent swing is 95% of the previous swing.

Thus, there is a G.P. that shows this situation,

Having first term, a = 15,

And, the common difference, r = 95 % = 0.95,

a) Also, for the 6th swing,

Number of terms, n = 6,

Hence, the distance covered by the pendulum on its 6th swing,

[tex]S_{n}=\frac{a(1-r^n)}{1-r}[/tex]

[tex]S_{6}=\frac{15(1-0.95^6)}{1-0.95}[/tex]

[tex]=79.4724328125\approx 79.47\text{ meters}[/tex]

b) When [tex]n=\infty[/tex]

The distance will the pendulum swing before it essentially stops is,

[tex]S_{\infty}=\frac{a}{1-r}[/tex]

[tex]=\frac{15}{1-0.95}[/tex]

[tex]=300\text{ meters}[/tex]

Answer:

a: z = -1.936

b: 0.0265

d: z < -1.645

Reject H0 if z < -1.645

Step-by-step explanation:

We are given:

H0:  µ = 20

HA:  µ < 20

n = 60, sample mean: 19.6, σ = 1.6

Since the alternate hypothesis has a < sign in it, it is a left tailed test.  The < or > sign in the alternate hypothesis points towards the rejection region.

For a:  We need to calculate the test statistic for our situation.  This is done with a z-score formula for samples.  

For b:  we need to use the z-score table to look up the p-value for the score we calculate in part a.  The p-value is 0.0265.  This means that there is only about a 2.65% chance that the sample values were a result of random chance.

For d:  Since the significance level is 0.05, and this is a one tailed test, we have a critical value of z < - 1.645.  This means that if the z-score we calculate in part a is less than -1.645, we will reject the null hypothesis

See attached photo for all the calculations!

The test statistic is approximately -4.84 with a p-value close to 0. The critical value is approximately -1.645. The rejection rule is to reject H0 if z < -1.645.

a. To compute the test statistic, we can use the formula: z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Given x = 19.6, μ = 20, σ = 1.6, and n = 60, we can calculate the test statistic as follows:

z = (19.6 - 20) / (1.6 / √60) = -1 / 0.2062 ≈ -4.84

b. The p-value can be determined by looking up the test statistic in the standard normal distribution table. However, in this case, the test statistic is already given as -4.84. The p-value associated with this test statistic is extremely small (close to 0), indicating strong evidence against the null hypothesis.

d. For a significance level α = 0.05, the critical value can be obtained by finding the z-score that corresponds to a cumulative probability of 1 - α = 0.95. Looking up this value in the standard normal distribution table, we find that the critical value is approximately -1.645.

The rejection rule using the critical value is: Reject H0 if z < -1.645.

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Answer:

the answer is a

Step-by-step explanation:

5x7= 35, 5x8= 40    40-35=5  

Answer:Michelle paid $56.15 for her shoes

Step-by-step explanation:

Since Michelle paid $3.89 more for her shoes compared to Conrad, who paid $52.26. This means you have to add $3.89 to $52.26.

$3.89+$52.25= $56.15

Final answer:

Michelle paid $56.15 for her shoes, which is $3.89 more than the $52.26 that Conrad paid for his.

Explanation:

The question involves a simple arithmetic calculation to determine how much Michelle paid for her shoes. To find the answer, we will add $3.89, which is the additional amount Michelle paid, to $52.26, which is the amount Conrad paid for his shoes.

Step-by-step calculation:

Therefore, Michelle paid $56.15 for her shoes.

Ans: It's C

'Cause (2/3 * 3/4x)+(2/3 * (-3/2)) = 1/2x - 1.

The expression which isequivalent to (1/2)x - 1 will be 2/3(3/4x - 3/2). Then the correct option is C.

The equivalent is the expression that is in different forms but is equal to the same value.

The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.

Let's check all the options, then we have

a) 1/3(3/2x - 1), simplify the expression, then we have

⇒ 1/3(3/2x - 1)

⇒ (1/2)x - 1/3

b) (3/2x + 1) - (x - 2), simplify the expression, then we have

⇒ (3/2x + 1) - (x - 2)

⇒ (1/2)x + 3

c) 2/3(3/4x - 3/2), simplify the expression, then we have

⇒ 2/3(3/4x - 3/2)

⇒ (1/2)x - 1

d) (3/4x - 2) + (1/4x + 1), simplify the expression, then we have

⇒ (3/4x - 2) + (1/4x + 1)

⇒ x - 1

The expression which isequivalent to (1/2)x - 1 will be 2/3(3/4x - 3/2). Then the correct option is C.

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Answer:

The answer is 28ft.

Step-by-step explanation:

A radius will always be exactly half of the diameter: meaning, the diameter is double the length of the radius.

Answer:

-9 C

Step-by-step explanation:

Subtract 4 from -5 will equal -9