The height, s, of a ball thrown straight down with initial speed 64 ft/sec from a cliff 80 feet high is s(t) = -16t2 - 64t + 80, where t is the time elapsed that the ball is in the air. What is the instantaneous velocity of the ball when it hits the ground? (2 points)
Select one:
a. 256 ft/sec
b. -96 ft/sec
c. 0 ft/sec
d. 112 ft/sec



The surface area of a right circular cylinder of height 4 feet and radius r feet is given by S(r)=2πrh+2πr2. Find the instantaneous rate of change of the surface area with respect to the radius, r, when r = 4. (2 points)
Select one:
a. 24π
b. 16π
c. 64π
d. 20π

The common speed of the blocks just after the collision is 0.4 times the initial velocity of the first block (v₁).

To determine the common speed of the blocks just after the completely inelastic collision, we can use the principle of conservation of linear momentum.

According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

Let's denote the velocity of the first block (m = 0.050 kg) before the collision as v₁ and the velocity of the second block (m = 0.075 kg) before the collision as v₂. Since the second block is stationary, its initial velocity, v₂, is 0.

1. Before the collision, the total momentum is given by:

Initial momentum = m₁v₁ + m₂v₂

where m₁ and m₂ are the masses of the first and second blocks, respectively.

Plugging in the values, we have:

Initial momentum = (0.050 kg) v₁ + (0.075 kg) 0

Initial momentum = (0.050 kg)v₁

2. After the collision, the two blocks stick together and move as a single unit with a common velocity, denoted as vf.

The final momentum is given by:

Final momentum = (m₁ + m₂) vf

where (m₁ + m₂) is the total mass of the system.

Plugging in the values, we have:

Final momentum = (0.050 kg + 0.075 kg) × vf

Final momentum = 0.125 kg × vf

According to the principle of conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision:

0.050 kg × v₁ = 0.125 kg × vf

To determine the common speed of the blocks just after the collision, we need to solve for vf:

vf = (0.050 kg × v₁) / (0.125 kg)

Calculating the numerical value, vf = 0.4 ×  v1

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Jose bought a bag of [tex]6[/tex] oranges for [tex]\$2.82[/tex] and Jose also bought [tex]5[/tex]  pineapples each cost [tex]\$3.15[/tex].

Explanation:

Given: Jose bought a bag of [tex]6[/tex] oranges for [tex]\$2.82[/tex]. He also bought [tex]5[/tex] pineapples. He gave the cashier [tex]\$20[/tex] and received [tex]\$1.43[/tex] change.

Let cost of a pineapple be [tex]x[/tex].

According to the question:

Cost of pineapple is calculated as [tex]20-(2.82+5x)=1.43[/tex]

                                                            [tex]20-2.82-5x=1.43\\[/tex]

                                                                               [tex]5x=17.18-1.43\\[/tex]

                                                                                 [tex]x=\frac{15.75}{5} \\x=3.15[/tex]

Therefore, cost of each pineapple is [tex]\$3.15[/tex].

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The scientific notation of the decimal number 0.0013 is [tex]0.0013 \times 10^{-3}[/tex].

Scientific notation is a way to express numbers in a concise form, particularly useful for very large or very small numbers.

It consists of two parts: a coefficient and an exponent of 10.

In the case of 0.0013, we can convert it to scientific notation as follows:

[tex]0.0013 \times 10^{-3}[/tex].

Hence, the scientific notation of 0.0013 is [tex]0.0013 \times 10^{-3}[/tex].

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The added zeros when changing a number from scientific notation to expanded notation are usually not considered significant. They primarily serve as placeholders and do not indicate precision of measurements.

In the field of Mathematics, specifically when working with numbers in scientific notation and expanded notation, the concept of significant zeros comes into play. When you convert a number from scientific notation to expanded notation, zeros that are included in this process are not considered to be significant. Rather, the concept of significant zeros mostly applies to measurements and the precision of those measurements. For example, in the measurement 300.0, the zero at the end is significant because it shows the measurement was precise to the tenth's place. However, in changing a number from scientific to expanded notation, any added zeros primarily serve as placeholders and are typically not considered significant.

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The inverse is f⁻¹(x) = -3(x + 1)/(x + 2).

To find the inverse of the given function f(x), follow these steps:

Thus, the inverse function is f⁻¹(x) = -3(x + 1)/(x + 2).

Answer:

Yes

No

Step-by-step explanation:

It is appropriate to eliminate parentheses as a first step. This can be done using the distributive property, resulting in either of ...

or by dividing the equation by the factor outside parentheses, resulting in ...

___

Any of the equations other than these represent violations of the equal sign. Something has been done to one side of the equation without the same thing being done to the other side.

Priscilla and Herb would take approximately 44.44 hours to address 10,000 envelopes together, as Priscilla can address 100 envelopes per hour and Herb can address 125 envelopes per hour.

The question asks about the combined work rate of Priscilla and Herb in addressing 10,000 envelopes. To solve this, we calculate each of their rates of work and then combine them to find the total rate. Priscilla completes 10,000 envelopes in 100 hours, so her rate is 100 envelopes per hour. Herb completes 10,000 envelopes in 80 hours, so his rate is 125 envelopes per hour. Working together, their combined rate is 100 + 125 = 225 envelopes per hour. To address 10,000 envelopes at this rate, we divide 10,000 by 225, which equals approximately 44.44 hours.

708 millimeters of juice Mrs. Goldstein pour into the bowl.

Given that, Mrs. Goldstein pours 3 juice boxes into a bowl to make fruit punch. Each juice box holds 236 millimeters.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the quantity of juice Mrs. Goldstein pour into the bowl be x millimeters.

Now, x=236 × 3

⇒ x = 708 millimeters

Therefore, 708 millimeters of juice Mrs. Goldstein pour into the bowl.

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To solve for the differential equation where y is the independent variable, one must interchange x and y, do some algebraic manipulations to separate the variables, and then integrate. Additional steps may be necessary to solve for individual constants in the equation, using given points such as x=2 when y=4. Given the absence of a starting function, a definite solution cannot be provided.

To find the solution of the differential equation (6−8xy2)dy/dx=y3 where y is the independent variable, we must first formulate the differential equation such that y is the independent variable. We do this by interchanging x and y, leading to (6-8x/y^2)dx/dy= x^3. By separating the variables and integrating, we will get the function y=f(x).

However, often with differential equations, a specific solution is also required. In this case, we know that x=2 when y=4. Plugging these into the function y=f(x) we can solve for the individual constant. Since no function f(x) is given in the question as a basis, it's impossible to come up with a definite solution, but I hope this general approach helps clarify how to solve such problems.

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Final answer:

The real numbers a and b that make the equation a+bi=13+9i true are a = 13 and b = 9. We match real and imaginary parts of the complex numbers to find a and b.

Explanation:

To find the real numbers a and b such that the equation a+bi=13+9i is true, we simply match the real parts and the imaginary parts of the complex numbers on both sides of the equation. The real part of a+bi is a, and the real part of 13+9i is 13. Therefore, a = 13. Similarly, the imaginary part of a+bi is bi and the imaginary part of 13+9i is 9i. Thus, b = 9. Hence, the values we are looking for are a = 13 and b = 9.

Answer:

Domain of h(x) : [2,∞)

Step-by-step explanation:

Given: Two functions

[tex]f(x)=\sqrt{2-x}[/tex]

[tex]g(x)=x-1[/tex]

Composite function, [tex]h(x)=\dfrac{f(x)}{g(x)}[/tex]

[tex]f(x)=\sqrt{2-x}[/tex], It is square root function. As we know square root function is always greater than 0.

2-x ≥ 0

  2 ≥ x

Domain of f(x) : [2,∞)

[tex]g(x)=x-1[/tex], It is straight line equation (Linear function) As we know domain of linear function is all real number.

Domain of g(x) : (-∞,∞)

[tex]h(x)=\dfrac{f(x)}{g(x)}[/tex]

[tex]h(x)=\dfrac{\sqrt{2-x}}{x-1}[/tex] , It is rational function. Denominator can't be zero.

So, x-1 ≠ 0

        x ≠ 1

Now, we will see domain of h(x) common all three domain.

Domain of h(x): [2,∞)

Hence, The domain of h(x) is [2,∞)

Given: Angle 1 and angle 2 are supplementary.

Angle 3 and angle 4 are supplementary.

Angle 1 is congruent to angle 3.

Prove: Angle 2 is congruent to angle 4.