Race times for a recent 5k
run are normally distributed with a mean
time of 38 minutes and standard devia-
tion of 3.5 minutes.
(a) What is the median race time for this
event?

Answer:

75

Step-by-step explanation:

1 minute=15

2 minutes=30

3 minutes=45

4 minutes=60

5 minutes=75

The surface area of a cube that has a width of 25 centimeters is 3750 cm².

It should be noted that surface area simply implies the measure of the total area that the surface of an object occupies.

In this case, the surface area of a cube is measured as 6a² where a = side

In this case, the side is 25. Therefore, the surface area will be:

= 6a²

= 6 × 25²

= 3750 cm²

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If A is located at (3,4) and B is located at (7,6) , then the slope of BC is -2  .

In the question ,

it is given that ,

In the rectangle ABCD , the coordinate of the point A is (3,4) and the coordinate of the point B is (7,6) .

since the sides of the rectangle are at 90° , that means they are perpendicular .

which means AB⊥BC ,

the slope of AB = (6-4)/(7-3)

slope = 2/4

slope = 1/2

Since , AB⊥BC ,

(slope of AB)×(slope of BC) = -1

substituting the values , we get

(1/2)×(slope of BC) = -1

slope of BC = -1/(1/2)

slope of BC = -2 .

Therefore , If A is located at (3,4) and B is located at (7,6) , then the slope of BC is -2  .

The given question is incomplete , the complete question is

On rectangle ABCD, if A is located at (3,4) and B is located at (7, 6), what is the slope of BC ?

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The probability of more than 5 people using the emergency room is 0.4729

Since 49% of the people in a community use the emergency room at a hospital in one year, a sample of 11 people use the emergency room. To find the probability of more than 5 people use the emergency room, we use binomial probability.

We find that the binomial probability of x is   P(X = x) = ⁿCₓpˣ(1 - p)ⁿ⁻ˣ

where

Since

Let us now find the probability of x ≤ 5.

So, P(X ≤ 5) = ¹¹C₀p⁰(1 - p)¹¹⁻⁰ + ¹¹C₁p¹(1 - p)¹¹⁻¹ + ¹¹C₂p²(1 - p)¹¹⁻² + ¹¹C₃p³(1 - p)¹¹⁻³

+ ¹¹C₄p²(1 - p)¹¹⁻⁴ + ¹¹C₅p⁵(1 - p)¹¹⁻⁵

So, substituting the values of the variables into the equation, we have

= ¹¹C₀(0.49)⁰(1 - 0.49)¹¹ + ¹¹C₁(0.49)¹(1 - 0.49)¹⁰ + ¹¹C₂(0.49)²(1 - 0.49)⁹ + ¹¹C₃(0.49)³(1 - 0.49)⁸

+ ¹¹C₄(0.49)⁴(1 - p)⁷ + ¹¹C₅(0.49)⁵(1 - 0.49)⁶

= (1)(1)(0.51)¹¹ + (11)(0.49)(0.51)¹⁰ + (55)(0.49)²(0.51)⁹ + (165)(0.49)³(0.51)⁸

+ (330)(0.49)⁴(0.51)⁷ + (462)(0.49)⁵(0.51)⁶

= 0.0006071 + (11)(0.49)(0.0011904) + (55)(0.2401)(0.0023341) + (165)(0.117649)(0.0045768)

+ (330)(0.0576480)(0.0089741) + (462)(0.0282475)(0.0175963)

= 0.0006071 + 0.0064163 + 0.0308230 + 0.0888452

+ 0.1707218 + 0.2296378

= 0.5270512

≅ 0.5271

So, P(X ≤ 5) = 0.5271, we see that the binomial probability of more than 5 people is P(X ≥ 5) = 1 - P(X ≤ 5)

So, substituting the value of the variable into the equation, we have

= 1 - 0.5271

= 0.4729

The probability is 0.4729

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The Isosceles obtuse triangle in which the obtuse vertex is angle T is represented with the attached diagram.

An isosceles obtuse triangle is a type of triangle which is formed by two equal sides of the triangle.

As long as the measures of the angles add up to 180°, it is an isosceles obtuse triangle.

An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.

A triangle is a polygon with three sides and three angles. There are different types of triangle, which includes:

So therefore, an isosceles obtuse triangle is formed by having two equal sides of a triangle.

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The given sequence160,80,40,20 is the geometric sequence.

In an Arithmetic sequence, the difference between the two consecutive terms is equal.

In a Geometric sequence, the common ratio of the two consecutive terms is equal.

The given sequence is 160, 80, 40, 20

Now,

[tex]T_{1} = 160\\T_{2} = 80\\T_{3} = 40\\T_{4} =20[/tex]

Subtracting the consecutive terms to calculate the common difference,

[tex]d = T_{2} -T_{1} = 80 -160 = -80[/tex]

[tex]T_{3} -T_{4} = 40 -80 = -40[/tex]

So, [tex]T_{2} -T_{1} \neq T_{3} -T_{4}[/tex]

So, the given sequence is not an arithmetic sequence.

Now, Calculating the common ratios,

[tex]r = \frac{T_{2} }{T_{1} } = \frac{80}{160} = \frac{1}{2}[/tex]

and, [tex]r =\frac{T_{3} }{T_{2} } = \frac{40}{80} = \frac{1}{2}[/tex]

Hence, the ratio is the same for consecutive terms.

So, the given sequence is the geometric sequence.

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[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &810000\\ r=rate\to 11.3\%\to \frac{11.3}{100}\dotfill &0.113\\ t=\textit{years}\dotfill &5\\ \end{cases} \\\\\\ A=810000(1 + 0.113)^{5} \implies A=810000(1.113)^5\implies A \approx 1383441.63[/tex]

Answer:

Step-by-step explanation:

2

Answer:

The leading coefficient is 2.

Step-by-step explanation:

Leading coefficients are the numbers with the highest degree or exponent.

Michael was assigned a total of 98 task which a fraction of 2/7 is 28

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

Data;

We can find the number of problem as

28 = 2/7 * x

7 * 28 = 2x

x = 98

The number of task assigned is 98.

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The scalene right triangle RGH is given by the image shown at the end of the answer.

There are two classifications of triangles, either regarding the side lengths or the angles.

Regarding the side lengths, the classifications are given as follows:

Regarding the measure of the greatest angle, the classifications are given as follows:

In this problem, RGH is a scalene right triangle, hence:

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The 0.566 meter, 0.87 meter, 10.7 rev/s and 7.7 rev/s radii and respective rotational speeds of young David's slings indicates.

a) The larger linear speed is 42.09 m/s

b) The centripetal acceleration of the sling at 7.7 rev/s is approximately 2,036.39 m/s²

A linear speed is a measure of the rate at which the distance traveled by an object in rotary motion changes with time.

When the length of the sling = 0.566 m, The rate of revolving of the sling = 10.7 rev/s

When the length of sling = 0.87 m, the rate of revolving the sling = 7.7 rev/s

Linear speed = Angular speed ×Radius length

The angular speed when the length of the is 0.566 m, ω₁, is found as follows;

One complete 360° rotation  = 2·π radians

ω₁ = 10.7 rev/s × 2·π ≈ 67.23 rad/s

The linear speed is; v₁ ≈ 67.23 rad/s × 0.566 m ≈ 38.05 m/s

The angular speed when the length is increased to 0.87 meters is therefore;

ω₂ = 7.7 rev/s × 2 × π ≈ 43.38 rad/sec

The larger of the two linear speed is approximately 42.09 m/s

Part 2: The centripetal acceleration, [tex]a_c[/tex], is found using the following formula;

[tex]a_c = \dfrac{v^2}{r}[/tex]

When the radius is r = 0.87 meters, the linear speed v₂ ≈ 42.09 m/s, which indicates;

[tex]a_c=\dfrac{42.09^2}{0.87} \approx 2,036.39[/tex]

The centripetal acceleration is approximately [tex]a_c[/tex] ≈ 2,036.39 m/s²

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Answer:

$55.71

Step-by-step explanation:

$320/7=$45.71 per day

45.71/8=5.71 per hour

45.71+10=$55.71 for 9 hours of work

45.71+ 45.71+55.71 =$147.13 for 3days +1 hour overtime

The ratio of dollars to paintbrushes is 7:4

A ratio expresses the number of times one value is contained in another value.

The number of paintbrushes Jane bought for $7 is 4

From the question, 4 paintbrushes can be bought with $7 (7 Dollars), therefore, seven dollars gives 4 paintbrushes

The ratio of dollars to paintbrushes is therefore 7 to 4, while the ratio of paintbrushes to dollars is 4 to 7

The expression for ratio is the colon symbol ':'

Ratios can also be expressed using the word 'to'.

A ratio can also be expressed using the hyphen '-'

The ratio of paintbrushes to dollars can be expressed as 4:7 or 4 to 7 or 4-7.

The correct option is therefore;

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Answer:

The ratio of paintbrushes to dollars is 4 to 7.

 The ratio of dollars to paintbrushes is 7:4.

 The ratio of paintbrushes to dollars is 4:7.

Step-by-step explanation:

it just is

If the given equation is y-25=-0.92(x-4). The equation in the slope-intercept form is y=-0.92x+28.68.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

The given equation is,y-25=-0.92(x-4)

The slope-intercept form of the equation is,

y=mx+c

m is the slope and c is the intercept.

Thus if the given equation is y-25=-0.92(x-4). The equation in the slope-intercept form is y=-0.92x+28.68.

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Answer:

12 [tex]\frac{1}{4}[/tex]

Step-by-step explanation:

substitute the given values for p and q into the expression

p² + 3q²

= (- [tex]\frac{1}{2}[/tex] )² + 3(- 2)²

= [tex]\frac{1}{4}[/tex] + 3(4)

= [tex]\frac{1}{4}[/tex] + 12

= 12 [tex]\frac{1}{4}[/tex]

A disadvantage of using a questionnaire is that B. The respondent controls the validity.

Questionnaires are a means of gathering data in research in order to find out more about a certain phenomenon. They involve writing questions and giving them to a respondent to fill out the answers.

Questionnaires are inexpensive to administer and can lead to confidential results which is good. However, they have the disadvantage that respondents control the validity of questionnaire results and can decide to lie if they feel it would make them look better, which would confuse the research results.

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The percent of families would be expected to have children who are all boys is 6.25%

The following can be illustrated:

all 4 boys BBBB 1 1/16

3 boys 1 girl BBBG,GBBB,BGBB,BBGB 4 4/16

2 boys 2 girl BBGG,BGBG,GBGB,BGGB,GBBG, GGBB 6 6/16

1 boy 3 grisl BGGG,GBGG,GGBG,GGGB 4 4/16

all 4 girls GGGG 1 1/16

Therefore, the percent of families would be expected to have children who are all boys will be:

= All boys × 100

= (1/16)*100

= 6.25%

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Yes, it is possible for a number to be a product of 2, 5, and 10.

All multiples of 10 such as 10, 20, 30, 40, 50, 60, ,,,, are a product of 2, 5, and 10.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

A number that is a product of 2, 5, and 10.

Example:

10 = 2 x 5

10 = 5 x 2

10 = 10 x 1

20

20 = 2 x 10

20 = 5 x 4

20 = 10 x 2

100

100 = 2 x 50

100 = 5 x 20

100 = 10 x 10

We see that all multiples of 10 are a product of 2, 5, and 10.

Thus,

All multiples of 10 are a product of 2,5, and 10.

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Answer:

4.25 minutes

Step-by-step explanation:

[tex]\frac{32 \:copies}{1\:min} = \frac{136\: copies}{x \: min} \\\\x\: min = \frac{136 \: copies}{32 \: copies} \\\\x=4.25 \: min[/tex]

Answer: 4 mins 15 seconds (4.25 mins)

Step-by-step explanation:

136 copies/32 copies per minute = 4.25 minutes

So we know it will take 4.25 mins. 4.25 mins is equal to 4 mins and 15 seconds.

After 2 hours the plan needs to be changed, and purchase solar for the cost effect.

It is defined as the conversion from one quantity unit to another quantity unit followed by the process of division, and multiplication by a conversion factor.

It is given that:

At 6 AM today, you purchased 1 MW of electricity contract for 12 PM at a price of 100 pounds/MWh.

Total cost = 6x100 = 600 pounds

Total cost for 2 hours = 200 pounds

Two hours later, the forecast for solar generation for 12 PM has changed from 4 GW to 4.5 GW.

Total cost for this:

From 8 AM to 12 PM = Number of hours 4 hours

4.5 GW = 4500 MW

= 105 pounds/MWh

Total cost for  4 hours = 630 pounds for 4500 MW per hour

Thus, after 2 hours the plan needs to be changed, and purchase solar for the cost effect.

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Total time taken by Bethany to make a round trip of NY to Pittsburgh as per  given conditions is equal to 1 (7/ 20)hours.

As given in the question,

Time taken by Bethany to flew from NY to Pittsburgh is equal to 3/4 hours

Now using jet stream ,

Time taken by airlines from Pittsburgh to NY is (4/5) th of (3/4) hours

= (4/5) × (3/4) hours

= 3/5 hours

Total time taken by airlines to make a round trip from NY to Pittsburgh is given by:

= [(3/4) + (3/5)] hours

= [(15 +12)/20] hours

= 27/20 hours

= 1 7/20 hours

Therefore, total time taken by Bethany to make a round trip of NY to Pittsburgh as per  given conditions is equal to  1 (7/ 20)hours.

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The solution of the composite function (f/g)(x) is [tex]\frac{-4x+3}{6+7x}[/tex]

Composite functions are when the output of one function is used as the input of another.

In other words, a composite function is generally a function that is written inside another function.

Composite function involves the composition of functions is combining two or more functions as a single function.

Therefore, let's solve the function below:

f(x) = -4x + 3

g(x) = 6 + 7x

Hence,

(f/g)(x) = f(x) / g(x)

Therefore,

f(x) / g(x) = [tex]\frac{-4x+3}{6+7x}[/tex]

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=========================================================

Explanation:

The three points are at these locations:

The notation "d(A,C)" means "the distance from A to C". It's equivalent to saying "the length of segment AC".

Then writing [tex]\left[d(A,C)]^2[/tex] means we'll square that distance.

Use the distance formula to get...

[tex]A = (x_1,y_1) = (10,6) \text{ and } C = (x_2, y_2) = (-5,3)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(10-(-5))^2 + (6-3)^2}\\\\d = \sqrt{(10+5)^2 + (6-3)^2}\\\\d = \sqrt{(15)^2 + (3)^2}\\\\d = \sqrt{225 + 9}\\\\d = \sqrt{234}\\\\[/tex]

This is the exact length of segment AC. That value squares to 234.

[tex]d = \sqrt{234} \ \to \ d^2 = (\sqrt{234})^2 = 234\\\\[/tex]

The square root and squaring operation cancel each other out. Think of it like fire vs water.

So we really only care about what's under the square root; rather than the entire square root expression itself. Which is nice because we don't have to worry about pesky things like decimal values.

This is why 234 is typed into the first box.

---------------------

Next, use the distance formula to find how far it is from A to B. Square the result to get what you see below.

[tex]A = (x_1,y_1) = (10,6) \text{ and } B = (x_2, y_2) = (1,-3)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(10-1)^2 + (6-(-3))^2}\\\\d = \sqrt{(10-1)^2 + (6+3)^2}\\\\d = \sqrt{(9)^2 + (9)^2}\\\\d = \sqrt{81 + 81}\\\\d = \sqrt{162}\\\\d^2 = (\sqrt{162})^2\\\\d^2 = 162\\\\[/tex]

This is the value of [tex]\left[d(A,B)\right]^2[/tex]

Now find the distance from B to C, and square the result.

[tex]B = (x_1,y_1) = (1,-3) \text{ and } C = (x_2, y_2) = (-5,3)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(1-(-5))^2 + (-3-3)^2}\\\\d = \sqrt{(1+5)^2 + (-3-3)^2}\\\\d = \sqrt{(6)^2 + (-6)^2}\\\\d = \sqrt{36 + 36}\\\\d = \sqrt{72}\\\\d^2 = \left(\sqrt{72}\right)^2\\\\d^2 = 72\\\\[/tex]

Add this to the previous squared value and we get 162+72 = 234, which matches exactly with the 234 found up toward the top.

We'll write 234 in the 2nd box as well.

This shows that [tex]\left[d(A,C)\right]^2 = \left[d(A,B)\right]^2+\left[d(B,C)\right]^2[/tex] is a true statement. By the converse of the Pythagorean theorem, we have confirmed this is a right triangle.

In other words, we've shown that [tex]a^2+b^2 = c^2[/tex] is a true statement (a,b,c are the sides of the right triangle such that c is the hypotenuse).

---------------------

Recall that we found these segment lengths:

[tex]AB = \sqrt{162} = \text{leg1}\\\\BC = \sqrt{72} = \text{leg2}\\\\AC = \sqrt{234} = \text{hypotenuse}\\\\[/tex]

The legs of a right triangle represent the base and height, in either order. This is because the legs are perpendicular to one another. They form a right (aka 90 degree) angle.

[tex]\text{area} = \frac{1}{2}*\text{base}*\text{height}\\\\\text{area} = \frac{1}{2}*\text{AB}*\text{BC}\\\\\text{area} = \frac{1}{2}*\sqrt{162}*\sqrt{72}\\\\\text{area} = \frac{1}{2}*\sqrt{162*72}\\\\\text{area} = \frac{1}{2}*\sqrt{11664}\\\\\text{area} = \frac{1}{2}*108\\\\\text{area} = 54\\\\[/tex]

Here are some alternative methods you can follow to find the area of this triangle.

As for verifying the answers, you can use a tool like GeoGebra.

Answer:

[tex][d(A,C)]^2=\boxed{234}[/tex]

[tex][d(A,B)]^2+[d(B,C)]^2=\boxed{234}[/tex]

[tex]\sf Area=\boxed{54}\; units^2[/tex]

Step-by-step explanation:

From inspection of the given diagram:

If ΔABC is a right triangle, the sum of the squares of the two shorter sides will equal the square of the longest side.  This is the definition of Pythagoras Theorem.

Use the distance formula to find the length of each side of the triangle.

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance between two points}\\\\$d=\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.\\\end{minipage}}[/tex]

[tex]\begin{aligned}d[(A,C)]&=\sqrt{(x_C-x_A)^2+(y_C-y_A)^2}\\&=\sqrt{(-5-10)^2+(3-6)^2}\\&=\sqrt{(-15)^2+(-3)^2}\\&=\sqrt{225+9}\\&=\sqrt{234}\end{aligned}[/tex]

[tex]\begin{aligned}d[(A,B)]&=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}\\&=\sqrt{(1-10)^2+(-3-6)^2}\\&=\sqrt{(9)^2+(-9)^2}\\&=\sqrt{81+81}\\&=\sqrt{162}\end{aligned}[/tex]

[tex]\begin{aligned}d[(B,C)]&=\sqrt{(x_C-x_B)^2+(y_C-y_B)^2}\\&=\sqrt{(-5-1)^2+(3-(-3))^2}\\&=\sqrt{(-6)^2+(6)^2}\\&=\sqrt{36+36}\\&=\sqrt{72}\end{aligned}[/tex]

Therefore:

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

Therefore, the triangle is a right triangle if:

[tex][d(A,B)]^2+[d(B,C)]^2=[d(A,C)]^2[/tex]

Substitute the found side lengths into the formula:

[tex]\implies [\sqrt{162}]^2+[\sqrt{72}]^2=[\sqrt{234}]^2[/tex]

[tex]\implies162+72=234[/tex]

[tex]\implies 234=234[/tex]

Hence proving that ΔABC is a right triangle.

To find the area of a right triangle, half the product of the two shorter sides:

[tex]\begin{aligned}\implies \sf Area &= \dfrac{1}{2}bh\\&=\dfrac{1}{2} \cdot [d(A,B)] \cdot [d(B,C)]\\&=\dfrac{1}{2} \cdot \sqrt{162} \cdot \sqrt{72}\\&=\dfrac{1}{2} \cdot \sqrt{162 \cdot 72}\\&=\dfrac{1}{2} \cdot \sqrt{11664}\\&=\dfrac{1}{2} \cdot \sqrt{108^2}\\&=\dfrac{1}{2} \cdot 108\\&=54 \sf \; units^2\end{aligned}[/tex]

The linear equation shown in the graph is the one in the third option.

y + 2 = -3*(x - 1)

Here we have a graph, and there we can see a linear equation, on that graph, we can see that:

The y-intercept is y = 1.

For each horizontal unit, the line goes 3 units down, then the slope is -3.

Then the linear equation is:

y = -3*x + 1

If now we add 2 in both sides, we get:

y + 2 = -3x +1 + 2

y + 2 = -3x + 3

y + 2 = -3*(x - 1)

So the correct option is the third one.

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By Computing the Logarithmic Relation, we calculate that the time between injections is 5.1 hours or 5 hours and 6 minutes.

The given formula is

[tex]D = 5e^{-0.1h}[/tex]

where D is the amount of drug in milligrams in the patient's bloodstream and h is the number of hours after the drug was administered.

So, it is given that D = 3 and we need to evaluate the number of hours it took between the injections,

So, We have,

[tex]D = 5e^{-0.1h}[/tex]

[tex]\frac{D}{5} = e^{-0.1h}[/tex]

Taking Natural log on both sides, we have :

[tex]ln\frac{D}{5} =ln e^{-0.1h}\\\\ln\frac{D}{5} = -0.1h\\[/tex]

We have D = 3, putting it in the formula, we get

[tex]\ln\frac{3}{5} = -0.1h\\\\ln(0.6) = -0.1h\\\\-0.5108 = -0.1 h\\\\h = \frac{0.5108}{0.1} = 5.1[/tex]

Hence, the time between injections is 5.1 hours or 5 hours and 6 minutes.

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Answer:

They should pay AED 180 as interest.

Step-by-step explanation:

Simple interest:

        P = The amount borrowed = AED 3000

        R = rate of interest = 3% per year

       t = time = 24 months = 24 ÷ 12 = 2 years

          [tex]\sf \boxed{Interest = PRt}[/tex]

Substitute the values,

                            [tex]\sf = 3000 * \dfrac{3}{100}*2\\\\ = 30*3*2\\\\=AED \ 180[/tex]

Answer: e) 6.7

Step-by-step explanation:

This is a Similarity of Triangles problem (geometry)

if x is the height of the lamp post, then:

x/5 = (3 + 9) / 9

x (9) = (12) (9)

x = 60/9 = 6.7