Nolan drops a ball from a certain height and allows the ball to bounce. The rebound height can be modeled by the function f(x) = 12(1/3)x. What is the initial height?

The total charges for both auto repair shops will be the same when 3 hours of labor have been performed.

Let's call the number of hours worked "h". The total charges for the first auto repair shop will be $30 (for the diagnosis) plus $25 per hour, or $30 + $25h. The total charges for the second auto repair shop will be $35 per hour, or $35h. We want to know when the total charges for both shops will be the same, so we can set the two equations equal to each other and solve for h:
$30 + $25h = $35h
$10h = $30
h = 3

So the total charges for both auto repair shops will be the same when 3 hours of labor have been performed.

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The similar triangles in increasing order are ΔEHF ~ ΔFHG ~ ΔEFG

The value of x = 13.5

Two triangles are similar if and only if:

Therefore, from the diagram, we can conclude that the similar triangles in increasing order of size is given as:

ΔEHF ~ ΔFHG ~ ΔEFG

Using the altitude rule, we will have the following proportion:

6/9 = 9/x

6x = 81

x = 13.5

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Answer: h=−2+i√107,−2−i√107

Answer:

75mm

Step-by-step explanation:

1cm = 10mm
u multiply 7.5 by 10

Answer:75

Step-by-step explanation:

No, the point (1,4) is not a solution of the set of linear inequalities y > 5x + 1 and y ≥ (1/2)x - 1 since the point does not satisfy both inequalities.

To determine if a point is a solution, we can substitute the x and y values of the point into the inequalities and see if they are true.

For the first inequality, y > 5x + 1:
4 > 5(1) + 1
4 > 6
This is not true, since 4 is not greater than 6. So the point (1,4) is not a solution for the first inequality.

For the second inequality, y ≥ (1/2)x - 1:
4 ≥ (1/2)(1) - 1
4 ≥ 0.5 - 1

4 ≥ -0.5

This is true, since 4 is greater than -0.5.

But since the point does not satisfy both inequalities, it is not a solution for the set of linear inequalities.

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Of these 12 individuals: Expected number of survivors ≈ 4.

Based on the given information, the chance of survival for the 12 people admitted to the hospital who tested positive for the disease is 30%.

If the chance of survival for a person who tests positive for the disease is 30%, and 12 people have been admitted and tested positive for the disease, then the expected number of survivors is 30% of 12, which is 3.6.

However, since it is not possible for 0.6 of a person to survive, we can round this number to the nearest whole number, which is 4.

Therefore, we can expect that 4 out of the 12 people who tested positive for the disease will survive. This can be written as:

Expected number of survivors = (Chance of survival) × (Number of people who tested positive) = (0.30) × (12) = 3.6 ≈ 4
So, the answer is 4.

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Answer:

$211 if rounding to nearest whole number (this may be what looking for)

$211.20 if rounding to nearest tenth

Step-by-step explanation:

The answer depends on what precision of rounding is done so i am providing 2 answers

Rounding to the nearest whole number:
14,49 rounded = 14
68.64 rounded = 69
128.05 rounded = 128

$14.49 + $68.64 + $128.05 =  14 + 69 + 128

= $211

Rounding to the tenths:

14,49 rounded = 14.5

68.64 rounded = 68.6

128.05 rounded = 128.1

$14.49 + $68.64 + $128.05= 14.5 + 68.6 + 128.1 = $211.20

Answer:

She earns $2,560

Step-by-step explanation:

She earns 2,560 because this is 8% of 32000. How do we know?
Multiply 32000 by 0.088, which is 8% of 100.

The final simplified expression is (3x+1)(5x+2)/(3x-2).

The given expression is (15x^(2)+13x+2)/(3x-2). We can try to simplify this expression by factoring the numerator and denominator, and then canceling out any common factors.

First, let's factor the numerator:

15x^(2)+13x+2 = (3x+1)(5x+2)

Now, let's factor the denominator:

3x-2 = (3x-2)

There are no common factors between the numerator and denominator, so we cannot simplify the expression any further.

Therefore, the simplified expression is:

(15x^(2)+13x+2)/(3x-2) = (3x+1)(5x+2)/(3x-2)

Since x!=(2)/(3), we do not need to worry about any undefined values.

So, the final simplified expression is:

(3x+1)(5x+2)/(3x-2)

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We have the following response after answering the given question: To interest the nearest penny, the balance in the account after 12 years is thus $17,821.76.

To calculate simple interest, divide the principal by the interest rate, the duration, and other variables. The marketing formula is simple return = capital + interest + hours. The easiest way to calculate interest is with this approach. The most popular method for calculating interest is as a percentage of the principal amount. For instance, if he borrows $100 from a friend and agrees to pay it back at 5% interest, he will only pay his portion of the 100% interest. $100 (0.05) = $5. Interest must be paid when you borrow money and must be added to any loans you make. The yearly percentage of the loan amount is frequently used to calculate interest. This percentage represents the loan's interest rate.

Continuously compounded interest is calculated as follows:

[tex]V = Pe^(rt) (rt)[/tex]

where: V = the investment's final value

P is the original investment's principle.

r equals the yearly interest rate (as a decimal)

t is the duration of the investment, in years.

P = $8290, r = 0.06 (6% as a decimal), and t = 12 years in this example.

So, [tex]V = 8290e^(0.06*12) = $17,821.76[/tex]

To the nearest penny, the balance in the account after 12 years is thus $17,821.76.

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The expression written using each base only once is 4¹¹ , the correct option is (a).

The expression 4⁸×4³ can be simplified using the rule of exponents,

The rule of exponents states that when we multiplying two exponential expressions with the same base, the exponents gets added.

which means that, nᵃ×nᵇ = nᵃ⁺ᵇ;

In this case the expression is: 4⁸×4³ , it has common base as "4",

So, by the rule of exponents, the power(exponents) gets added up ;

The expression "4⁸×4³" can be rewritten as 4⁸⁺³, which is equal to 4¹¹,

Therefore, Option(a)4¹¹, is the expression written using each base only once.

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The given question is incomplete, the complete question is

What is the expression written using each base only once? 4⁸×4³

(a) 4¹¹

(b) 12¹¹

(c) 4²⁴

(d) 64¹¹.

The value of a when b=4 and c=4 is 72.70. Rounded to two decimal places, the answer is 72.70.

The variable a is jointly proportional to the cube of b and the square of c. This means that the relationship between a, b, and c can be expressed as: a = k * b^3 * c^2, where k is a constant of proportionality.

When a=433, b=5, and c=7, we can plug these values into the equation to find the value of k:
433 = k * 5^3 * 7^2
433 = k * 125 * 49
433 = k * 6125
k = 433/6125
k = 0.07069

Now, we can use this value of k to find the value of a when b=4 and c=4:
a = k * b^3 * c^2
a = 0.07069 * 4^3 * 4^2
a = 0.07069 * 64 * 16
a = 72.70

Therefore, the value of a when b=4 and c=4 is 72.70. Rounded to two decimal places is 72.70.

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Answer:

chatGPT

Step-by-step explanation:

Let's denote the speed of each car as v, and the distance that Car A travels as d. Then we can set up two equations based on the information given:

d = v * (12/60) (since Car A reaches its destination in 12 minutes)

d + 4 = v * (18/60) (since Car B travels 4 miles farther than Car A and reaches its destination in 18 minutes)

Simplifying the equations by multiplying both sides by 60 (to convert the minutes to hours) and canceling out v, we get:

12v = 60d

18v = 60d + 240

Subtracting the first equation from the second, we get:

6v = 240

Therefore:

v = 240/6 = 40

So the cars travel at a speed of 40 miles per hour.

Answer:

597 x 89 = 53133.

53133 beads

Step-by-step explanation:

The value of a is 576.43, when a varies directly as b and inversely as the square of c.

Given that a varies directly as b and inversely as the square of c, we can write the equation as:

a = k * (b/c²)

Where k is the constant of proportionality.

We are given that a=113 when b=7 and c=8, so we can plug these values into the equation and solve for k:

113 = k * (7/8²)

113 = k * (7/64)

k = 113 * (64/7)

k = 1036.57

Now that we know the value of k, we can plug in the new values of b and c to find a:

a = 1036.57 * (5/3²)

a = 1036.57 * (5/9)

a = 576.43

Therefore, when b=5 and c=3, a=576.43.

Round your answer to two decimal places if necessary, so the final answer is:

a = 576.43

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Answer:

A perfect square trinomial is a trinomial expression of the form:

a^2 + 2ab + b^2

Where a and b are constants, it can also be written as (a + b)^2.

To identify a perfect square trinomial, we can check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms.

For example, let's consider the expression:

x^2 + 4x + 4

The first term is x^2, which is a perfect square. The last term is 4, which is also a perfect square. The middle term is 4x, twice the product of the square roots of x^2 and 4 (i.e., 2x). Therefore, this expression is a perfect square trinomial:

x^2 + 4x + 4 = (x + 2)^2

To simplify this expression, we can use the fact that (a + b)^2 = a^2 + 2ab + b^2:

(x + 2)^2 = x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4

Therefore, the perfect square trinomial x^2 + 4x + 4 is equivalent to (x + 2)^2.

Step-by-step explanation:

Based on the graph of this linear function, the domain and range are as follows;

Domain = {0, 100}.

Range = {450, 1200}.

In Mathematics, a domain is the set of all real numbers for which a particular function is defined.

Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = {0, 100}.

Range = {450, 1200}.

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It will take 18 years for the Type A and Type B trees to be the same height.

A location in the centre of a line connecting two places is referred to as the midpoint. The midpoint of a line is located between the two reference points, which are its ends. The line that connects these two places is split in half equally at the halfway. In addition, the halfway is reached if a line is drawn to divide the line that connects these two places.

Let us suppose the number of years = y.

The height of the tree of type A is thus,

5 feet + (4 inches/year)(y)

= 5 + 4/12 y feet

= 5 + 1/3 y feet

For the type B:

2 feet + (10 inches/year)(y years)

= 2 + 10/12 y feet

= 2 + 5/6 y feet

To get the height of the two trees as equal we have:

5 + 1/3 y = 2 + 5/6 y

3 + 1/3 y = 5/6 y

3 = 1/6 y

18 = y

Hence, it will take 18 years for the Type A and Type B trees to be the same height.

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Answer:

1. around 4.86 or 4.9 for the nearest tenth

2. 17 months

Step-by-step explanation:

1. (1.7×10^6)/(3.5×10^5) = (1.7/3.5)×(10^6/10^5) = 0.4857×10 = 4.857

Therefore, 1.7×10^6 is about 4.857 times as great as 3.5×10^5.

2.  start by finding out how much Erica still owes after the down payment:

Total cost - Down payment = $1,867 - $320 = $1,547

divide the amount still owed by the monthly payment to find out how many months Erica will be paying:

$1,547 ÷ $91 per month = 17 months (rounded up)

Therefore, Erica will be paying for the bike for 17 months

The number of liters in each container is 8.24 liters.

Division is a mathematical operation, in which we distribute the number in equal parts, the number on the upper side is the total quantity and the number on the bottom side is equal parts of numbers which have to be distributed.

We denote division by '÷' this symbol.

Destiny combined a total of 8.27 + 6.65 = 14.92 liters of paint to make purple paint.

She poured this paint equally into 2 containers, which means each container has half of the total amount of paint:

= 14.92 / 2

= 7.46 liters

However, Destiny also had 1.56 liters of paint left over that she didn't use.

To divide the remaining paint equally between the two containers, each container gets half of the remaining paint:

= 1.56 / 2

= 0.78 liters

Therefore, each container has 7.46 + 0.78 = 8.24 liters of paint.

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The volume of the right circular cylinder is approximately 14,864.77 cubic meters.

What is the right circular cylinder?

A right circular cylinder is a three-dimensional solid shape that consists of a circular base and a curved side that is perpendicular to the base. The term "right" refers to the fact that the axis of the cylinder is perpendicular to the base, and the term "circular" refers to the shape of the base, which is a circle.

The formula for the volume of a right circular cylinder is V = πr^2h, where r is the radius of the base and h is the height of the cylinder. The formula for the surface area of a right circular cylinder is A = 2πrh + 2πr^2, where r is the radius of the base and h is the height of the cylinder.

Right circular cylinders are commonly used in engineering, architecture, and everyday objects such as cans, pipes, and containers.

The formula for the volume of a right circular cylinder is [tex]V = \pi r^2h[/tex], where r is the radius of the base and h is the height of the cylinder.

In this case, the radius is 17.2 meters and the height is 15.3 meters.

Plugging in these values, we get:

[tex]V = \pi (17.2)^2(15.3)[/tex]

V ≈ 14,864.77 cubic meters

Therefore, the volume of the right circular cylinder is approximately 14,864.77 cubic meters.

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Answer:

See the attached worksheet.

Step-by-step explanation:

This assumes that the second "Beans" entry was supposed to be "Carrots, instead.

Note how the equations match the describtion in the paragraph.  The multiply the only portion of the equation not explained is the (3y+5) expression in both equations.  By process of elimination (marked), this must be the area of the garden.

Since we are not given an actual numeric value for the area, the only thing we can write in Part B is to duplicate the simplified equations from Part A.

The solution to the inequality is option C. x > -9.

Linear inequalities are defined as those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠ and the value of the exponent of the variable is 1.

The given inequality is,

-1/3x + 1/2 < 3.5

We have to find the solution of the inequality.

-1/3 x + 0.5 < 3.5

Subtracting both sides by 0.5, we get,

-1/3 x < 3.5 - 0.5

-1/3 x < 3

Multiplying both sides by -3,

x > -9

Hence the solution is x > -9.

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LHS ≠ RHS and the equation is not an identity.

9. Simplify each expression:
a. √(1 + cos 76° / 2)
b. (sin 158.2°) / (1 + cos 158.2°)
10. Verify that each equation is an identity.
a. sin 2x / 2 sin x = cos^2 x/2 – sin^2 x/2
b. tan θ/2 = csc θ – cot θ

Answer:
9. a. √(1 + cos 76° / 2) = √(1 + 0.2419 / 2) = √(1 + 0.12095) = √(1.12095) = 1.0589
b. (sin 158.2°) / (1 + cos 158.2°) = (0.12088) / (1 + (-0.9927)) = 0.12088 / 0.0073 = 16.566
10. a. sin 2x / 2 sin x = cos^2 x/2 – sin^2 x/2
LHS = sin 2x / 2 sin x = 2 sin x cos x / 2 sin x = cos x
RHS = cos^2 x/2 – sin^2 x/2 = (cos x + sin x)(cos x - sin x) / 4 = (1)(cos x - sin x) / 4 = cos x - sin x / 4
Therefore, LHS ≠ RHS and the equation is not an identity.
b. tan θ/2 = csc θ – cot θ
LHS = tan θ/2 = sin θ/2 / cos θ/2
RHS = csc θ – cot θ = 1 / sin θ - 1 / cos θ = (cos θ - sin θ) / (sin θ cos θ)
Therefore, LHS ≠ RHS and the equation is not an identity.

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The equation of the curve that passes through the point (1, 8) and has the gradient function 3x² + 5x⁴ is y = x³ + x⁵ + 6

In this problem, we are given the gradient function 3x² + 5x⁴ and the point (1, 8) through which the curve passes. To find the equation of the curve, we need to integrate the gradient function and add the constant of integration. Let's start by integrating 3x² + 5x⁴:

∫(3x² + 5x⁴)dx = x³ + x⁵ + C

where C is the constant of integration.

Now we have the general equation of the curve, but we need to find the specific value of C that will give us the curve that passes through the point (1, 8). We can do this by plugging in the values of x and y into the equation:

8 = 1³ + 1⁵ + C

8 = 1 + 1 + C

C = 6

Therefore, the equation of the curve is:

y = x³ + x⁵ + 6

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f(x)=3x+5

g(x)=1/(x-3)

Q1 The point-slope equation of the line passing through the points (5,10) and (-3,12) is y-10=2(x-5) and the slope-intercept equation of the same line is y=2x-9.
Q2 a) The equation of the line which passes through (8,0) and is parallel to the line with equation y=2x-6 is y=2x.
b) The equation of the line which passes through (12,3) and is perpendicular to the line with equation y=2x-6 is y=-1/2x+9.
c) The equation of the line which passes through the points (1,2) and (5,6) is y=2x-1.
Q3 a) (f+g)(x) = 3x+5 + 1/(x-3)
b) (f⋅g)(x) = 3x+5 * 1/(x-3)
c) (2f+3g)(x) = 6x+10 + 3/(x-3)
d) (3g-4)(x) = 3/(x-3) - 4
Q4 a) (f2g)(x) = (x2)2/(x+1)
b) (∘f min g)(x) = x/(x+1)
Q5 The domain of the function f(x)=3x+5 and g(x)=1/(x-3) is all real numbers except 3 and the range of both the functions is all real numbers.

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Answer: 15

Step-by-step explanation:

Answer:

[tex]\boxed{n= 15}[/tex]

Step-by-step explanation:

we can use the following formula:

[tex]\alpha = \frac{180(n-2)}{n}[/tex]

This formula helps to calculate the sum of the interior angles of a polygon, where:

we have the value of the interior angle, and we need "n", so we will solve for "n":

[tex]156= \frac{180(n-2)}{n}\\156n=\frac{180(n-2) \not{n}}{\not{n}}\\156n= 180n-360\\-24n=-360\\n= 15[/tex]

With this we have solved the exercise.

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

793 people in the country said that they like country music if 23% of the people surveyed prefer country music.

The given data is as follows:

percentage of people prefer music = 23%

People did not like country music = 2653

Let us assume make this equation has equal properties,

0.23x = people who like country music

We know that 2,653 people did not like music. We can write this equation as:

x - 0.23x = 2653

0.77x = 2653

x = 2565 / 0.77

x = 3446.75

Taking the x value approximately, we get the x value as,

x = 3447

Now we can substitute the x value in the above equation, we get,

0.23x = 0.23(3447)

= 792.81

Therefore we can conclude that 793 people in the country said that they like country music.

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The correct answer is option d) \( 1+\tan x \), since \( \sec x=1+\tan x \).

The correct answer for the given expression \( \tan x \cdot \sin x+\cos x=? \) is option d) \( 1+\tan x \).

Step-by-step explanation:

We can start by using the identity \( \tan x = \frac{\sin x}{\cos x} \) to rewrite the expression:
\( \frac{\sin x}{\cos x} \cdot \sin x+\cos x=? \)

Next, we can simplify the expression by multiplying the numerator and denominator by \( \cos x \):
\( \frac{\sin^2 x+\cos^2 x}{\cos x}=? \)

Now, we can use the identity \( \sin^2 x+\cos^2 x=1 \) to further simplify the expression:
\( \frac{1}{\cos x}=? \)

Finally, we can use the identity \( \frac{1}{\cos x}=\sec x \) to rewrite the expression in terms of \( \sec x \):
\( \sec x=? \)

Therefore, the correct answer is option d) \( 1+\tan x \), since \( \sec x=1+\tan x \).

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Answer:

1 = 128

2 = 52

3 = 52

4 = 128

5 = 128

6 = 52

7 = 52

8 = 128