Mr. Woods is buying ice cream for two of his students, Moses and Gaby. They each get one scoop of ice cream. Gaby wants ice cream in a cone, but she is a slow eater. Moses wants a cup because he is a bit messy and doesn’t want to drip any ice cream. Each scoop is a perfect sphere and has a radius of 5 cm. The height of the cup and cone are both 12 cm. and the radius of each container is the same as each scoop of ice cream.

The inverse of the function f(x) is f⁻¹(x) = √(x) - 8

To find the inverse of the function f(x) = (x + 8)², we need to solve for x in terms of y:

y = (x + 8)²

Taking the square root of both sides, we get:

±√(y) = x + 8

Solving for x, we get:

x = ±√(y) - 8

Since we want the inverse function to be a function (i.e., have a unique output for each input), we must choose the positive square root. Therefore, the inverse function is:

f⁻¹(x) = √(x) - 8

The domain of f⁻¹ is the range of f, which is [0, ∞). Therefore, the domain of f⁻¹ is [0, ∞).

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The tangent of 60° is √3, so the coterminal angle of tan(780°) is √3.

Coterminal angles are angles that have the same initial and terminal sides in standard position.

To find the coterminal angle of tan(780°), we need to add or subtract multiples of 360° to 780° until we get an angle between 0° and 360°, because angles that differ by a multiple of 360° have the same trigonometric function values.

First, we can subtract 360° from 780°:

780° - 360° = 420°

This is not yet between 0° and 360°, so we can subtract another 360°:

420° - 360° = 60°

Now we have an angle between 0° and 360° that is coterminal with 780°.

The tangent function has a period of 180°, which means that the tangent function has the same value for angles that differ by a multiple of 180°. Since 60° is an acute angle, we can use the tangent of 60° to find the tangent of 780°:

tan(780°) is equivalent to tan(780° - 720°) = tan(60°)

The tangent of 60° is √3, so the coterminal angle of tan(780°) is √3.

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The answer is 369,072. This is the total number of copies sold to date.

An expression that uses symbols to represent the relationship between two or more values.

This is calculated by solving the equation 9.7% × x = 35,800, where x is the total number of copies sold to date.

To solve this equation, we first need to convert the percentage to a decimal.

To do this, we divide 9.7 by 100, giving us 0.097.

We can then multiply both sides of the equation by this decimal, giving us 0.097x = 35,800.

We can then solve for x by dividing both sides of the equation by 0.097. This gives us x = 369,072. This is the total number of copies sold to date.

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The product of 3.18 × 16 is 50.88 in decimal form.

A number system is a way to represent numbers using symbols or digits. The most commonly used number system is the decimal or base-10 system, which uses ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent any quantity.

Other common number systems include:

Binary or base-2 system, which uses two digits (0 and 1) to represent numbers

Octal or base-8 system, which uses eight digits (0, 1, 2, 3, 4, 5, 6, 7) to represent numbers

Hexadecimal or base-16 system, which uses sixteen digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F) to represent numbers

Knowing that;

3.18 × 16

After multiplying, we now have;

3.18 × 16 = 50.88

Thus, the product of 3.18 × 16 is 50.88 in decimal form.

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Answer:

[tex]x = 4 \frac{2}{3} [/tex]

[tex]y = 5[/tex]

Step-by-step explanation:

Multiply the second equation by -5 to eliminate 15x:

{15x - 4y = -50,

{3x - 2y = -16; / × (-5)

+ {15x - 4y = -50,

{-15x + 10y = 80;

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

6y = 30 / : 6

y = 5

Make x the subject from the 2nd equation (it doesn't matter, you can do it from the 1st one instead):

15x = -50 + 4y / : 15

[tex]x = 3 \frac{1}{3} + \frac{4}{15} y[/tex]

[tex]x = 3 \frac{1}{3} + \frac{4}{15} \times 5 = \frac{14}{3} = 4 \frac{2}{3} [/tex]

The antiderivative of the given function is: [tex]-3/x^2 + 8ln|x| + C[/tex], where x is not equal to 0.

What is antiderivative?

Antiderivative is the reverse process of differentiation in calculus. Given a function f(x), an antiderivative of f(x) is another function F(x) whose derivative is equal to f(x).

The given function can be written as:

[tex](9x^3+8x^5)/x^6 = 9/x^3 + 8/x[/tex]

To find the antiderivative, we integrate each term separately using the power rule of integration:

∫(9/[tex]x^3[/tex]) dx = -3/[tex]x^2[/tex] + C1

and

∫(8/x) dx = 8ln|x| + C2

where C1 and C2 are constants of integration.

Therefore, the antiderivative of the given function is:

∫([tex]9x^3+8x^5[/tex])/[tex]x^6 dx[/tex] = ∫([tex]9/x^3[/tex]) dx + ∫(8/x) dx

= ([tex]-3/x^2 + C1[/tex]) + (8ln|x| + C2)

= [tex]-3/x^2[/tex] + 8ln|x| + C

where C = C1 + C2 is a constant of integration. Therefore, the antiderivative of the given function is:

[tex]-3/x^2 + 8ln|x| + C[/tex], where x is not equal to 0.

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Two events are dependent since the probability of getting a perfect square is affected by the fact that we already know that we got an even number.

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain. For example, the probability of rolling a 6 on a fair dice is 1/6, or approximately 0.167.

In probability theory, events are often expressed as sets of possible outcomes, and probabilities are calculated based on the number of outcomes in the event relative to the total number of possible outcomes. For example, the probability of rolling an even number on a fair dice is 3/6, or 0.5, because there are three even numbers (2, 4, and 6) out of a total of six possible outcomes.

Since the event E is the event of getting an even number, the possible outcomes are 2, 4, 6, or 8.

Out of these, only 4 is a perfect square. Therefore, P(S/E) = 1/4.

These two events are dependent since the probability of getting a perfect square is affected by the fact that we already know that we got an even number.

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Answer:

Got you bro

Step-by-step explanation:

The two-dimensional shape appears to be a semi-circle, and it is being rotated about a line to form a three-dimensional shape. The resulting shape is a sphere.

Answer:

Step-by-step explanation:

The two-dimensional shape appears to be a semi-circle, and it is being rotated about a line to form a three-dimensional shape.

The resulting shape is a sphere.

The product of 3 and 10v√40 in simplest radical form is 30v(2√5).

What is product?

To find the product of 3 and 10v√40 in simplest radical form, we can simplify the radical first.

First, we can simplify 40 by finding its prime factorization:

40 = 2 × 2 × 2 × 5

Next, we can rewrite 10v√40 as 10v√(2 × 2 × 2 × 5) to separate out the perfect squares:

10v√(2 × 2 × 2 × 5) = 10v(√2 × √2 × √2 × √5)

We can then simplify the perfect squares under the radical:

10v(√2 × √2 × √2 × √5) = 10v(2√5)

Now we can multiply 3 and 10v(2√5):

3 × 10v(2√5) = 30v(2√5)

So the product of 3 and 10v√40 in simplest radical form is 30v(2√5).

What is prime factorization?

Prime factorization is the process of expressing a composite number as a product of its prime factors. In other words, it is finding the prime numbers that can be multiplied together to get the original number. For example, the prime factorization of 24 is 2 x 2 x 2 x 3 or 2³ x 3, since 24 can be expressed as a product of the prime numbers 2 and 3, and each of these primes is repeated as many times as necessary to get the original number. Prime factorization is an important concept in mathematics and has many practical applications, including in cryptography, number theory, and computer science.

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[tex]\textit{Law of sines} \\\\ \cfrac{\sin(\measuredangle A)}{a}=\cfrac{\sin(\measuredangle B)}{b}=\cfrac{\sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin(B)}{7}=\cfrac{\sin(30^o)}{6}\implies \sin(B)=\cfrac{7\sin(30^o)}{6} \\\\\\ B=\sin^{-1}\left[ \cfrac{7\sin(30^o)}{6} \right]\implies B\approx 35.69^o[/tex]

Make sure your calculator is in Degree mode.

The probability that a conversation lasts longer than 6 minutes is approximately 0.0764 or 7.64%.

What is probability?

The probability of an event is a figure that represents how probable it is that the event will take place. In terms of percentage notation, it is stated as a number between 0 and 1, or between 0% and 100%. The higher the probability, the more probable it is that the event will take place.

We can use the standard normal distribution to solve this problem by standardizing the variable using the formula:

z = (x - μ) / σ

where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

In this case, we want to find the probability that a conversation lasts longer than 6 minutes, so x = 6, μ = 5, and σ = 0.7. Substituting these values into the formula, we get:

z = (6 - 5) / 0.7 = 1.43

Next, we can use a standard normal distribution table or a calculator to find the probability that a standard normal random variable is greater than 1.43. Using a standard normal distribution table, we find that this probability is approximately 0.0764.

Therefore, the probability that a conversation lasts longer than 6 minutes is approximately 0.0764 or 7.64%.

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Rolle’s Theorem can be applied to the closed interval and the value of x = (12 ±√12)/3

Rolle's theorem states that "If a function f is defined in the closed interval [a, b] in such a way that it satisfies the following condition: If is continuous οn [a, b], ii) f is differentiable οn (a, b), and iii) f (a) = f (b), then there exists at leastοne value οf x, let us assume this value to be c, which lies between a and b i.e. (a < c < b) in such a way that f'(c) = 0.".

Here, we have

Given: f(x) = (x-1)(x-2)(x-3), [1,3]

We have to determine whether Rolle’s Theorem can be applied to the closed interval.

This function is continuous in [1, 3] and is differentiable everywhere except at the points x = 1, 2, 3.

This point is in the interval [1, 3], and since Rolle's Theorem requires that the function must be differentiable on the open interval (1, 3).

f(x) = (x-1)(x-2)(x-3)

f'(x) = (x-2)(x-3) + (x-1)(x-3) + (x-1)(x-2)

f'(x) = x² - 5x + 6 + x² - 4x + 3 + x² -3x + 2

f'(x) = 3x² -12x + 11

f'(x) = 0

3x² -12x + 11 = 0

x = (12 ±√12)/3

Hence, Rolle’s Theorem can be applied to the closed interval.

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Therefore, the absolute maximum value of f(x) over (-2, 3) is 201 and it occurs at x = -2. The absolute minimum value of f(x) over (-2, 3)is -98 and it occurs at x = 3.

We must identify the crucial points of the function within the period in order to determine the function's absolute maximum and minimum values.

Define critical point?

The critical points are those where the function's derivative is zero or undefinable. The function is then assessed at these pivotal points as well as the interval's endpoints.

Absolute maximum and lowest values are represented by the largest and smallest values, respectively.

The derivative of f(x) = 3x⁴ - 4x³ - 12x² + 1 over (-2, 3) is initially found as follows:

f'(x) = 12x³ - 12x²- 24x

If we set f'(x) to 0, we obtain:

12x³ - 12x² - 24x = 0

By multiplying both sides of this equation by 12x, we may simplify it to:

x²- x - 2 = 0

The answer to this quadratic equation is:

x = -1, x = 0, x = 2

Now we evaluate f(x) at these critical points and at the endpoints of the interval:

f(-2) = 201

f(-1) = -6

f(0) = 1

f(2) = 49

f(3) = -98

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Answer: For x < -2.5:

y = |2x + 5| - |2x - 5|

y = -(2x + 5) - (-(2x - 5)) (since 2x - 5 < 0 and 2x + 5 < 0 for x < -2.5)

y = -2x - 5 + 2x - 5

y = -10

For x > 2.5:

y = |2x + 5| - |2x - 5|

y = 2x + 5 - (2x - 5) (since 2x - 5 < 0 and 2x + 5 > 0 for x > 2.5)

y = 10

For -2.5 ≤ x ≤ 2.5:

y = |2x + 5| - |2x - 5|

y = 2x + 5 - (-(2x - 5)) (since 2x - 5 < 0 and 2x + 5 > 0 for -2.5 ≤ x ≤ 2.5)

y = 4x

Step-by-step explanation:

Answer:

The slope of a line represents the rate of change, which in this case is the rate at which the skydiver is falling. The rate of falling is 4,800 feet in 4 seconds, which simplifies to 1,200 feet per second, therefore, the graph that has a slope of 1,200 best represents this rate.

Answer:A complex number will, in general, have five(5) fifth complex roots./

Step-by-step explanation:

The correct answer of the following is option A which is 49+32+m =180 because an angle on a straight line is equal to 180 degree.

what is angle?

Angles are an important concept in mathematics, physics, and engineering. An angle is formed by two rays or lines that share a common endpoint called a vertex. The measurement of an angle is usually expressed in degrees or radians.

In geometry, angles are classified according to their measure. An acute angle measures less than 90 degrees, a right angle measures exactly 90 degrees, an obtuse angle measures greater than 90 degrees but less than 180 degrees, and a straight angle measures exactly 180 degrees.

Angles can be formed by lines that intersect or by shapes such as triangles, quadrilaterals, and circles. The study of angles is an important part of trigonometry, which is the branch of mathematics concerned with the relationships between angles and the sides and heights of triangles.

Angles are also used in physics to describe the orientation and motion of objects. For example, the angle between two vectors can be used to calculate the direction of a force or the trajectory of a moving object.

In engineering, angles are used to design structures such as bridges, buildings, and machines. Engineers use trigonometry to calculate the angles and lengths of the various components of these structures to ensure they are safe and structurally sound.

Overall, angles are a fundamental concept in mathematics and have numerous applications in science, technology, and everyday life.

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Answer:

$67,000 after 15 years.

Step-by-step explanation:

You deposit $6000 in an account earning 7% interest compounded monthly. How much will you have in the account in 15 years? account will have $67,000 in it after 15 years

The width of the coal tray is approximately 16.24 inches.

The width of the coal tray is determined as follows:

The given formula is P α 1/(1 + d²).

We can rewrite it as P = k/(1 + d²), where k is a constant of proportionality.

Since the problem states that the width of the coal tray is equal to d, we can assume that the width of the tray is equal to 1 (arbitrary units), without loss of generality.

So, we have P = k/(1 + d²) = k/(1 + (distance from food to coals / 1)²)

P = k/(1 + distance from food to coals²)

When the food is 16 inches above the tray, the distance from food to coals is d = 16/1 = 16.

When P = 0.53, we have:

0.53 = k/(1 + 16²)

k = 0.53(1 + 16²)

k ≈ 139.88

Now we can use the equation P = 0.53 = 139.88/(1 + d²) to solve for d:

0.53(1 + d²) = 139.88

1 + d² = 264.45

d² = 263.45

d ≈ 16.24

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Answer:

9.6 / 0.91 ≈ 10.55

Step-by-step explanation:

I used a calculator

Answer:

Step-by-step explanation:

The sequence starts with 4 and each subsequent term is obtained by multiplying the previous term by -3. Therefore, the equation to describe the sequence is:

aₙ = 4(-3)^(n-1)

where aₙ represents the nth term in the sequence.

Using this equation, we can find the values of the first few terms in the sequence as follows:

a₁ = 4(-3)^(1-1) = 4(1) = 4

a₂ = 4(-3)^(2-1) = 4(-3) = -12

a₃ = 4(-3)^(3-1) = 4(9) = 36

and so on.

Note that the index n starts at 1, but in some contexts it may start at 0, in which case we would need to adjust the exponent accordingly.

Answer:

√3/2

Step-by-step explanation:

sin(10°) • cos(50°) + cos(10°) • sin(50°)

= sin(10 + 50)

sin(60°) = √3/2

The original price of the bracelet was £1400.

Percentage is a way of expressing a proportion or a fraction as a part of 100. It is denoted using the symbol "%". For example, 50% means 50 out of 100, or half, while 25% means 25 out of 100, or one-quarter.

We can start by using the information given to set up an equation that relates the original price of the bracelet with the sale price and the percentage reduction:

original price x (100% - 70%) = sale price

Simplifying the percentage reduction:

original price x 30% = sale price

Substituting the given sale price (£420):

original price x 30% = £420

To solve for the original price, we need to isolate it on one side of the equation. We can do this by dividing both sides by 30% (which is the same as multiplying by 100/30 or 10/3):

original price = sale price / 30% = £420 / 30% = £1400

Therefore, the original price of the bracelet was £1400.

Complete question - A jewelry shop is having a sale. A bracelet is now reduced to £420. This is 70% of the original price. Work out the original price of the bracelet.

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The required statement is "The sky is clear if and only if the plane is on time."

The statement "Q<->P" is a logical statement that uses the biconditional operator "<->" which means "if and only if." This operator connects two propositions in such a way that both propositions are true or false together.

In this case, the propositions are "Q" and "P" which are defined as "The sky is clear" and "The plane is on time," respectively. Therefore, the statement "Q<->P" can be translated into words as "The sky is clear if and only if the plane is on time."

This means that the truth of the proposition "Q" (the sky is clear) is dependent on the truth of proposition "P" (the plane is on time) and vice versa.

If the plane is on time, then the sky must be clear, and if the sky is clear, then the plane must be on time. If either of these propositions is false, then the bi-conditional statement is false as well.

Thus, required statement is "The sky is clear if and only if the plane is on time."

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The population will exceed 2458 after approximately 10.1465 units of time, where the time unit depends on the context of the problem (e.g., hours, days, etc.).

A statement proving the equality of two expressions is known as an equation. It can include variables, integers, and mathematical operations like addition, subtraction, multiplication, and division. It also incorporates mathematical symbols. In mathematics and science, equations are frequently used to illustrate connections between quantities. The equals sign (=) is typically used in equations to denote that the expressions on each side of the sign have the same value. For instance, the formula 2 + 3 = 5 demonstrates that the total of 2 and 3 equals 5. Equations can be solved to determine a variable's value or to determine if a certain value meets the connection that the equation describes.

To estimate when the population will exceed 2458, we can set up an inequality using the equation for the population growth:

p(t) > 2458

Substituting the given equation for p(t), we get:

[tex]1100e^0.12t[/tex] > 2458

Dividing both sides by 1100, we get:

[tex]e^0.12t > 2.23545[/tex]

Taking the natural logarithm of both sides, we get:

0.12t > ln(2.23545)

Solving for t, we get:

t > ln(2.23545)/0.12

Using a graphing calculator to evaluate this expression, we get:

t > 10.1465

Therefore, the population will exceed 2458 after approximately 10.1465 units of time, where the time unit depends on the context of the problem (e.g., hours, days, etc.).

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The balances and the running total, given the checking account record is :

The balance on the 14 th of February would be:

= Balance forward - Amount of check

= 299. 88 - 90. 48

= $ 209. 40

The balance on 2 / 15 :

= 209. 40 - 65

= $ 144. 40

The balance on 2 / 17 :

= 144. 40 + 381.33 which is a deposit

= $ 525.73

The balance and running total on 2 / 22 :

= 525. 73 - 54.47

= $ 471.26

The balance on 2 / 22 after the National Mortgage deduction is:

= 471. 26 - 375.99

= $ 95.27

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PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations.

PEMDAS rule states that the order of operation starts with the parentheses first or the calculation, which is enclosed in brackets. Then, the operation is performed on exponents(degree or square roots), and later, we do operations on multiplication & division and at last addition and subtraction.

PEMDAS is the order that you solve a problem.

PEMDAS stands for:

Parenthesis

Exponents

Multiplication and Division

Addition and Subtraction

If you have an equation or expression that you need to solve, you solve it in this order. You do the parenthesis first, then you solve the exponents, then you do the multiplication and division from left to right, and last you do addition and subtraction from left to right.

In the equation

2 + 3 * 5,

first you would do 3 * 5 because multiplication is before addition.

2 + 15

Then you would do the addition to get the answer:

17.

Please give brainliest

Answer: For Damian's investment:

The interest rate is 3%, compounded quarterly, which means the interest rate per quarter is 3%/4 = 0.75%.

The number of quarters in 16 years is 16*4 = 64.

Using the formula for compound interest, the balance after 16 years is:

A = P*(1 + r/n)^(n*t)

where:

P = the principal (initial investment) = $81,000

r = the interest rate per quarter = 0.75%

n = the number of times the interest is compounded per year = 4 (quarterly)

t = the number of years = 16

A = 81000*(1 + 0.0075/4)^(4*16) = $157,222.39

For Marques's investment:

The interest rate is 2%, compounded continuously.

Using the formula for continuous compound interest, the balance after 16 years is:

A = Pe^(rt)

where:

P = the principal (initial investment) = $81,000

r = the interest rate per year = 2%

t = the number of years = 16

A = 81000e^(0.0216) = $131,518.16

Therefore, Damian would have $157,222.39 - $131,518.16 = $25,704.23 more than Marques in his account after 16 years. Rounded to the nearest dollar, this is $25,704.

Step-by-step explanation:

Damian would have approximately $351 more in his account than Marques after 16 years.

Interest rate is the percentage of a loan or deposit that is charged as interest or earned as interest over a period of time. It is expressed as a percentage of the principal amount borrowed or deposited, and it represents the cost of borrowing or the reward for saving money.

We can use the compound interest formula to calculate the future value of each investment after 16 years and then subtract to find the difference.

For Damian's investment, the interest rate is 3% per year, compounded quarterly. This means that the quarterly interest rate is r = 0.03/4 = 0.0075, and the number of compounding periods is n = 16 x 4 = 64. The future value of Damian's investment is:

F = 81000 * [tex](1 + r)^n[/tex]

  = 81000 * [tex](1.0075)^64[/tex]  

  = 129,535.28

For Marques's investment, the interest rate is 2% per year, compounded continuously. This means that the continuously compounded interest rate is r = 0.02, and the number of compounding periods is n = 16 x 1 = 16. The future value of Marques's investment is:

F = 81000 * [tex]e^(rn)[/tex]

  = 81000 * [tex]e^(0.0216)[/tex]

  = 129,183.81

The difference between the two investments is:

129,535.28 - 129,183.81 = 351.47

So Damian would have approximately $351 more in his account than Marques after 16 years. Rounded to the nearest dollar, the difference is $351.

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Answer:

x = 3√3;

y = 3

Step-by-step explanation:

Use trigonometry:

[tex] \sin(30°) = \frac{y}{6} [/tex]

Cross-multiply to find y:

[tex]y = 6 \times \sin(30°) = 6 \times 0.5 = 3[/tex]

Use the Pythagorean theorem to find x:

[tex] {x}^{2} = {6}^{2} - {y}^{2} [/tex]

[tex] {x}^{2} = {6}^{2} - {3}^{2} = 36 - 9 = 27[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3} [/tex]