Which parent function has a constant slope?

A. linear
B. square root
C. quadratic
D. exponential

Answer: is a

Step-by-step explanation:

Final answer:

The slope of a line is calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. The formula represents the rise over run between these points.

Explanation:

We can use a specific formula to calculate the slope of a line using two points on that line. The slope is found by taking the difference in the y-value (also known as the rise) and dividing it by the difference in x-value (also known as the run). In mathematical terms, if we have two points (x1,y1) and (x2,y2), the formula for slope (m) is as follows:

m = (y2 - y1) / (x2 - x1)

Therefore, between two given points, you subtract the y-value of the starting point from the y-value of the end point to find the rise, and then remove the x-value of the starting point from the x-value of the end point to determine the run. The slope is then the quotient of these two differences.

Water flows: 34300 gallons in 7 minutes. Tank empties in about 3.914 minutes. (Rounded to three decimal places.)

To find out how much water flows out of the tank in the first 7 minutes, we need to integrate the flow rate function, [tex]\(f(t)\)[/tex], from [tex]\(t = 0\)[/tex] to [tex]\(t = 7\)[/tex]. The flow rate function is given as [tex]\(f(t) = 300t^2\)[/tex].

So, the amount of water that flows out of the tank in the first 7 minutes, denoted as [tex]\(W_{\text{out}}\)[/tex], is given by the integral of [tex]\(f(t)\)[/tex] over the interval [tex]\([0, 7]\)[/tex]:

[tex]\[W_{\text{out}} = \int_{0}^{7} f(t) \, dt\][/tex]

[tex]\[= \int_{0}^{7} 300t^2 \, dt\][/tex]

[tex]\[= 300 \int_{0}^{7} t^2 \, dt\][/tex]

To find the integral of [tex]\(t^2\)[/tex], we use the power rule for integration:

[tex]\[= 300 \left[\frac{t^3}{3}\right]_{0}^{7}\][/tex]

[tex]\[= 300 \left(\frac{7^3}{3} - \frac{0^3}{3}\right)\][/tex]

[tex]\[= 300 \left(\frac{343}{3}\right)\][/tex]

[tex]\[= 34300\][/tex]

So, [tex]\(W_{\text{out}} = 34300\)[/tex] gallons.

Now, to find out how many minutes it takes for the tank to be completely empty, we need to find the time, [tex]\(T\)[/tex], at which the tank is empty. The tank will be empty when the integral of the flow rate function from [tex]\(t = 0\)[/tex] to [tex]\(t = T\)[/tex] is equal to the total volume of the tank, which is 6000 gallons.

So, we have:

[tex]\[6000 = \int_{0}^{T} f(t) \, dt\][/tex]

[tex]\[= \int_{0}^{T} 300t^2 \, dt\][/tex]

[tex]\[= 300 \int_{0}^{T} t^2 \, dt\][/tex]

Using the power rule for integration again:

[tex]\[6000 = 300 \left[\frac{t^3}{3}\right]_{0}^{T}\][/tex]

[tex]\[= 300 \left(\frac{T^3}{3} - \frac{0^3}{3}\right)\][/tex]

[tex]\[= 100T^3\][/tex]

Now, solving for [tex]\(T\):[/tex]

[tex]\[T^3 = \frac{6000}{100}\][/tex]

[tex]\[T^3 = 60\][/tex]

[tex]\[T = \sqrt[3]{60}\][/tex]

[tex]\[T \approx 3.914\][/tex]

So, it takes approximately 3.914 minutes for the tank to be completely empty.

Answer:

5

Step-by-step explanation:

The t represents days, and we can't have a fraction of a day for the population to reach [tex]1,000,000[/tex] cells, we need to round up to the nearest whole number. Therefore, it will take approximately [tex]17[/tex] days for the population to reach [tex]1,000,000[/tex] cells.

To model the population of cells, we can use an exponential growth equation. Since the population doubles every day, the equation would be:

[tex]P(t)=P_{0} *2^{t}[/tex]

Where:

[tex]P(t)[/tex] is the population size at time t.

[tex]P_{0}[/tex] is the initial population size ([tex]10[/tex] cells in this case).

[tex]t[/tex] is the time in days.

So, for this scenario, the equation becomes:

[tex]P(t)=10*2^{t}[/tex]

Now, we want to find out how many days it will take for the population to reach [tex]1,000,000[/tex] cells. We can set up the equation:

[tex]1,000,000=10*2^{t}[/tex]

To solve for t, we need to isolate it. First, divide both sides by [tex]10[/tex] :

[tex]100,000=2 t[/tex]

Now, take the logarithm base 2 of both sides to solve for t:

[tex]log_{2} (100,000)=t[/tex]

Using a calculator, we find:

[tex]t=16.61[/tex]

Since t represents days, and we can't have a fraction of a day for the population to reach [tex]1,000,000[/tex] cells, we need to round up to the nearest whole number. Therefore, it will take approximately [tex]17[/tex] days for the population to reach [tex]1,000,000[/tex] cells.

COMPLETE QUESTION:

Create an equation to model a population of cells which starts with [tex]10[/tex]cells and doubles in size every day. How many days will it take to reach [tex]1,000,000[/tex] cells?

When two tangents are drawn to a circle at endpoints A and B of a diameter, the tangents are perpendicular to the radii at their points of tangency and are parallel to each other, forming two congruent right triangles.

The question involves constructing two tangents to a circle from two points, A and B, where AB is the diameter of the circle. By the properties of circles and tangents, we know that a tangent to a circle is perpendicular to the radius at the point of tangency. Since AB is the diameter, points A and B are on opposite ends of the circle, and when we draw tangents at these points, each tangent will be perpendicular to the radii OA and OB respectively. Additionally, since AB is a diameter, it creates a linear pair with the two tangents, meaning that the two tangents are parallel to each other. This setup creates two congruent right triangles, namely triangle OAT and triangle OBT, where T represents the points of tangency on each tangent.

Two tangents to a circle from points A and B, where AB is a diameter, are congruent and parallel to each other, forming right angles with the diameter.

When constructing two tangents to a circle at points A and B, where AB is a diameter, an interesting relationship exists between those tangents. By the properties of tangents to a circle from an external point, we know that two tangents drawn to a circle from the same external point are equal in length. Therefore, the two tangents you would draw from points A and B to any point on the circumference of the circle would be congruent to each other.

Additionally, since each tangent is perpendicular to the radius of the circle at the point of tangency, the tangents are parallel to each other because they are both perpendicular to the same diameter. In the context of circle O with AB as a diameter, we can also conclude that the angles formed between the tangents and the diameter AB are right angles due to the geometric properties of tangents.

The expression gives the area of the triangle shown below  1/2cx is the answer.

The reason this is the answer is because to find the area of a triangle you will find half of the base times the height.

The formula of a triangle

A=1/2bh.

In this picture, the base is the length of the bottom part of the triangle (c).

It is the distance of the solid line only.

The height is represented as the straight line that is perpendicular to the base (forms a right/90-degree angle).

X is the height of the triangle.

Therefore, 1/2cx is the answer.

So option D is correct.

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Answer: D.  Alternate interior angles of parallel lines cut by a transversal are congruent.

Step-by-step explanation:

In the given picture m is parallel  to CB  .

Since m is a straight line and the angle of straight line is 180°  .

Therefore, m∠XAB+m∠BAC+m∠YAC=180° ........................ (a)

Also, m is parallel to CB and AB is a transversal and because Alternate interior angles of parallel lines cut by a transversal are congruent.

⇒m∠XAB=m∠ABC  and  m∠YAC=m∠ACB

Put the values of m∠XAB and m∠YAC in equation (a), we get

m∠ABC + m∠BAC + m∠ACB = 180°.

Hence proved.

Statement D Alternate interior angles of parallel lines cut by a transversal are congruent is the most relevant to the congruence of XAB and ABC.

It is proved that  m∠ABC + m∠BAC + m∠ACB = 180°.

For angle XAB being congruent to angle ABC, and then prove the equation m∠ABC + m∠BAC + m∠ACB = 180°.

Parallel lines m and CB.

Points X, A, and Y are on line m.

To prove that angle XAB is congruent to angle ABC.

Justification for Congruence:

In this context, use the Alternate Interior Angles Theorem,

which states that when a transversal intersects two parallel lines, the alternate interior angles formed are congruent.

Here's the reasoning:

Line m is parallel to line CB.

The line XY is a transversal line that intersects m and CB.

Therefore, the alternate interior angles formed by this setup are congruent.

This means that angle XAB and angle ABC are congruent.

Now, let's prove the equation m∠ABC + m∠BAC + m∠ACB = 180°.

Angle XAB is congruent to angle ABC (justified by the Alternate Interior Angles Theorem).

Now, let's consider the triangle ABC. By the angle sum property of triangles, the sum of the angles in a triangle is always 180°. Therefore:

m∠ABC + m∠BAC + m∠ACB = 180°

Since angle XAB is congruent to angle ABC, replace angle ABC with angle XAB:

m∠XAB + m∠BAC + m∠ACB = 180°

Now, the equation holds true, and you have successfully proven that m∠ABC + m∠BAC + m∠ACB = 180°.

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Answer:

In a number cube, a dice, you have six sides with the numbers 1, 2, 3, 4, 5 and 6. And when you roll it, each number has the same probability of showing.

a) rolling a 3 twice.

In the first roll you get a 3, with a probability of 1/6, and with the second roll you also get a 3, also with a probability of 1/6. The probability of bot evets to happen is equal to the product of the individual probabilities.

P = 1/6*1/6 = 1/36

b) An even number and a 5.

In the first roll you can get 2, 4 or 6, so you have 3 out of 6 options, the probability is 3/6 = 1/2, in the second roll you need you get a 5, the probability is 1/6.

P = 1/2*1/6 = 1/12.

But we also have the probability where the first dice is a 5 and the second a pair number, so the real probability is 2*P = 1/6

c) an odd number and 2 or 4.

for an odd number the probability is 3/6 = 1/2, for getting 2 or 4 the probability is 2/6 = 1/3

then we have that P = 2*(1/3)*(1/2) = 1/3

d) a number less than 6, you have 5 options, then the probability is:

p = 5/6, and for 3 or 1 the probability is 2/6 = 1/3

then P = 2*(5/6)*(1/3) = 5/9

76 baseballs and 24 bats were ordered.

Let's assume that 'x' represents the number of baseballs ordered and 'y' represents the number of bats ordered.

We can set up two equations to solve for 'x' and 'y':

We can solve the system of equations using the substitution or elimination method. Let's use the elimination method by multiplying the first equation by 4.50:

Subtracting the first equation from the second equation eliminates 'x' and we can solve for 'y':

15.50y = 372

y = 24

Substituting the value of 'y' back into the first equation, we can solve for 'x':

x + 24 = 100

x = 100 - 24

x = 76

Therefore, 76 baseballs and 24 bats were ordered.

76 baseballs and 24 bats were ordered for the summer camp, costing $822 in total.

Let's denote the number of baseballs as [tex]\( B \)[/tex] and the number of bats as [tex]\( T \)[/tex].

We're given two pieces of information:

1. The total number of items ordered is 100, so [tex]\( B + T = 100 \)[/tex].

2. The total cost of the items is $822, so [tex]\( 4.50B + 20T = 822 \)[/tex].

We can solve this system of equations by substitution or elimination. Let's use the elimination method.

1. Multiply the first equation by 20 to match the coefficient of [tex]\( T \)[/tex] in the second equation:

[tex]\[ 20(B + T) = 20(100) \][/tex]

[tex]\[ 20B + 20T = 2000 \][/tex]

2. Subtract this modified first equation from the second equation:

[tex]\[ (4.50B + 20T) - (20B + 20T) = 822 - 2000 \][/tex]

[tex]\[ 4.50B + 20T - 20B - 20T = -1178 \][/tex]

[tex]\[ -15.50B = -1178 \][/tex]

3. Divide both sides by -15.50 to solve for [tex]\( B \)[/tex]:

[tex]\[ B = \frac{-1178}{-15.50} \][/tex]

[tex]\[ B \approx 76 \][/tex]

4. Now, substitute the value of [tex]\( B \)[/tex] back into the first equation to find [tex]\( T \)[/tex]:

[tex]\[ 76 + T = 100 \][/tex]

[tex]\[ T = 100 - 76 \][/tex]

[tex]\[ T = 24 \][/tex]

So, there were 76 baseballs and 24 bats ordered.

The correct answer is:

False.

Explanation:

When you take out a loan, banks provide all of the relevant information for the loan.  This includes the interest rate, payment amount, and an amortization schedule, which shows the amount paid and the amount of principal this takes off of the loan every month for the life of the loan.

Answer: 42

Step-by-step explanation:

x = 6 now that you know what x represents switch it out the x by the 7 with a 6 like this 7(x)- 7(6) seven times six is 42 in other words x

=42

The area of the sector has a diameter of 21.2 cm and a central angle of 3π/5 radians is 105.90 square cm.

It is a locus of a point drawn at an equidistant from the center. The distance from the center to the circumference is called the radius of the circle.

The central angle of 3π/5 radians and a diameter of 21.2 cm.

Then the area of the sector will be

[tex]\rm Area = \dfrac{\theta}{2\pi} *\dfrac{\pi}{4} d^2[/tex]

Then put the values, we have

[tex]\rm Area = \dfrac{\frac{3\pi}{5}}{2\pi} *\dfrac{\pi}{4} (21.2)^2\\\\\\\rm Area = \dfrac{3\pi}{10\pi} *\dfrac{3.14}{4} *449.44\\\\\\Area = \dfrac{3}{10} *\dfrac{3.14}{4} *449.44\\\\\\Area = 105.8968 \approx 105.90 \ cm^2[/tex]

The area of the sector has a diameter of 21.2 cm and a central angle of 3π/5 radians is 105.90 square cm.

More about the circle link is given below.

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Answer:

A. (–2, –3)

that is the answer

Final answer:

The expression tan5x - tan2y / 1 + tan5x tan2y simplifies to tan(5x + 2y) using the tangent sum formula, illustrating how two tangent values can be combined into a single tangent of the sum of two angles.

Explanation:

The expression given is tan5x - tan2y / 1 + tan5x tan2y. This can be simplified using the tangent sum formula, which is a key concept in trigonometry. The tangent sum formula states that tan(a + b) = (tan a + tan b) / (1 - tan a tan b).

By comparing the given expression to this formula, we can see that the expression represents the tangent of the sum of two angles, specifically tan(5x + 2y). This simplification uses trigonometric identities to represent the combination of two different tangent values as a single tangent value of the sum of the angles.

Answer:

2(3a + 6) would be correct.

Hope this was helpful !

Step-by-step explanation: