Find the (real-valued) general solution to the differential equation. z″+8z′=0 z(t)=

Answer:

282.6 inches cubed

Step-by-step explanation:

The volume of the cylinder with radius 3 in and height 10 in is

[tex]V=\pi (3^2)(10)= \pi (9)(10)= 90\pi = 90(3.14) = 282.6 in^3[/tex]

Answer:

90 in3

Step-by-step explanation:

Answer:

HI!

Your spinner has 5 colors, and if you spin it, the probability of landing in each color will be the same ( because the spiner has 5 equal sections).

So for every spin, the probability on landing on each color will be 20%.

If i spin it 200 times, then the 20% of 200 is 0.2*200 = 40.

It means that if you spin it 200 times, then each colour shows 40 times theoretically. The question is: how many times should Alina expect the spinner to land on either purple or orange?

you have 40 for purple and 40 for orange, then the total times that the spinner lands on either purple or orange is 80.

Answer:

C

Step-by-step explanation:

Answer:

6^1/20

Step-by-step explanation:

you multiply the two exponents

The correct order is [tex]l_n,\ m_n,\ t_n,\ r_n,\ i[/tex].

To list the values ln, rn, mn, tn and i in increasing order for any value of n, we need to understand the relationship between these variables.

The variables ln, rn, mn, and tn likely represent different measures, such as left endpoint, right endpoint, midpoint, and trapezoid point in numerical methods, while i often represents the imaginary unit iota in mathematics.

Commonly, if we assume these have increasing values when ordered, they would be listed as:

Therefore, the values in increasing order are:

[tex]l_n,\ m_n,\ t_n,\ r_n,\ i[/tex]

Answer:

[tex]\frac{4\sqrt{6x}}{2x}[/tex]

Explanation:

The problem we are given is

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

We can write the square root of a fraction as a fraction with a separate radical for the numerator and denominator; this gives us

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

We can write the whole number 4 as the fraction 4/1; this gives us

[tex]\frac{4}{1}\times \frac{\sqrt{3}}{\sqrt{2x}}\\\\=\frac{4\sqrt{3}}{\sqrt{2x}}[/tex]

We now need to "rationalize the denominator."  This means we need to cancel the square root in the denominator.  In order to do this, we multiply both numerator and denominator by √(2x); this is because squaring a square root will cancel it:

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

When multiplying radicals, we can extend the radical over both factors:

[tex]\frac{4\sqrt{3} \times \sqrt{2x}}{2x}\\\\=\frac{4\sqrt{3\times 2x}}{2x}\\\\=\frac{4\sqrt{6x}}{2x}[/tex]

Answer:

The correct answer is ^4sqrt24x^3/2x or B on edge.

Step-by-step explanation:

To find out how far the student can ride in 45 minutes, we calculate the speed of 0.15 miles per minute from the given 3 miles in 20 minutes, then multiply this speed by 45 minutes, resulting in a distance of 6.75 miles.

The student has stated that they can travel 3 miles in 20 minutes. To find out how far they can ride in 45 minutes, we need to calculate their riding speed and then apply it to the longer time period.

Step 1: Calculate the riding speed

The speed can be calculated by dividing the distance (3 miles) by the time (20 minutes), which gives us a speed of 0.15 miles per minute.

Step 2: Use the speed to determine the distance in 45 minutes

Now that we know the speed, we can multiply it by the new time period (45 minutes) to find the distance they can ride. That calculation is 0.15 miles/minute × 45 minutes = 6.75 miles.

When we check whether the answer is reasonable, we can notice that 45 minutes is a little more than twice 20 minutes, and since they can ride 3 miles in 20 minutes, it makes sense that they could ride more than twice that distance in 45 minutes, so our answer of 6.75 miles seems reasonable.

They can ride 6.75 miles in 45 minutes.

To find how far they can ride in 45 minutes, we can use the concept of average speed.

Given that they can ride 3 miles in 20 minutes, we can calculate their average speed as:

Average Speed = Distance / Time

Average Speed = 3 miles / 20 minutes

Average Speed = 0.15 miles per minute

Now, we can find how far they can ride in 45 minutes:

Distance = Average Speed x Time

Distance = 0.15 miles per minute x 45 minutes

Distance = 6.75 miles

Therefore, they can ride 6.75 miles in 45 minutes.

Final answer:

To find the weekly wage for someone working 40 hours at an hourly rate of $9.75, multiply the number of hours by the hourly rate, which equals $390.

Explanation:

The weekly wage for a person who works 40 hours at an hourly rate of $9.75 can be calculated by multiplying the number of hours worked per week by the hourly wage. Here's the calculation:

Therefore, the answer required for the weekly wage for this person is $390.

Final answer:

To maximize revenue from football game ticket sales and concessions, we create a revenue function based on ticket price adjustments and evaluate changes in quantity demanded. The optimal price and attendance are found by analyzing the combined revenue function, deriving it with respect to the number of price decreases, and finding the maximum value of this function.

Explanation:

To determine the ticket price that should be charged by the university to maximize revenue from football game attendance, we need to analyze the effect of price changes on the quantity demanded. We can create a revenue function based on the given figures and the assumptions that a decrease of $3 in ticket price increases attendance by 10,000 people, and each person spends an average of $4.50 on concessions.

Let x represent the number of $3 decreases in price from the original $18. The ticket price can then be expressed as P = 18 - 3x, and attendance as A = 40000 + 10000x. The revenue from ticket sales is R1 = P × A, which gives us R1 = (18 - 3x) × (40000 + 10000x). The concession revenue per person is $4.50, so total concession revenue is R2 = 4.50 × A, which gives us R2 = 4.50 × (40000 + 10000x).

To maximize total revenue R = R1 + R2, we need to find the maximum of the function R = (18 - 3x) × (40000 + 10000x) + 4.50 × (40000 + 10000x). To find this, we can use calculus to take the derivative of R with respect to x and find the value of x that makes this derivative zero. This will be the critical point which we can then test to confirm it provides a maximum.

A polynomial that is written as a product has been "factored completely."

They are mathematical expressions involving variables raised with non negative integers and coefficients(constants who are in multiplication with those variables) and constants with only operations of addition, subtraction, multiplication and non negative exponentiation of variables involved.

In the expression Terms are added or subtracted to make a polynomial. They're composed of variables and constants all in multiplication.

Since we know that factorization is the method of breaking a number into smaller numbers that multiplied together will give that original form.

A polynomial which can be written as a product has been "factored completely."

Learn more about polynomials here:

https://brainly.com/question/27343162

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Final answer:

The school district pays $42 less by hiring a substitute teacher for a day instead of paying Ms. Maple, calculating by comparing Ms. Maple's daily salary to a substitute teacher's daily rate.

Explanation:

To calculate how much less the school district pays by hiring a substitute teacher instead of paying Ms. Maple for a day, we first need to determine Ms. Maple’s daily salary. This is done by dividing her annual salary by the number of working days in the school year.

Therefore, the school district pays $42 less by hiring a substitute teacher for a day instead of paying Ms. Maple.

Final answer:

The given function f(x) = x3 – 5x2 + 9x – 45 has rational roots ±1, ±3, ±5, and ±9.

Explanation:

To determine the number of roots and identify them for the function f(x) = x3 – 5x2 + 9x – 45, we can use the Rational Root Theorem and synthetic division. The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation, then p must divide the constant term (in this case, 45) and q must divide the leading coefficient (in this case, 1). By trying all possible combinations of p and q, we can find the rational roots of the equation. In this case, the rational roots are ±1, ±3, ±5, and ±9.

The number of real roots is 1, and it is ( x = 5 ).

To determine the number of roots and identify them for the function[tex]\( f(x) = x^3 - 5x^2 + 9x - 45 \)[/tex], we can use various methods such as the Rational Root Theorem, Descartes' Rule of Signs, or graphing techniques. In this case, since the degree of the polynomial is 3, we know that there will be 3 roots in total.

We'll start by checking for rational roots using the Rational Root Theorem. According to this theorem, if a rational root [tex]\( \frac{p}{q} \)[/tex] exists for the polynomial [tex]\( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), then \( p \)[/tex] is a factor of the constant term [tex]\( a_0 \) and \( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex].

The constant term of [tex]\( f(x) \) is \( -45 \)[/tex]and the leading coefficient is ( 1 ). The factors of ( -45 ) are [tex]\( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 \)[/tex]. The factors of \( 1 \) are \( \pm 1 \)[tex]\( 1 \) are \( \pm 1 \)[/tex].

By trying all possible combinations of these factors, we can find the rational roots of the polynomial. We'll then use synthetic division or polynomial long division to check if these roots are actually roots of the polynomial.

Let's proceed with the calculations:

1. Possible rational roots:

[tex]\( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 \)[/tex]

2. Testing these roots using synthetic division or polynomial long division:

Testing [tex]\( x = 1 \)[/tex]:

[tex]\[ f(1) = (1)^3 - 5(1)^2 + 9(1) - 45 = 1 - 5 + 9 - 45 = -40 \][/tex]

Testing [tex]\( x = -1 \)[/tex]:

[tex]\[ f(-1) = (-1)^3 - 5(-1)^2 + 9(-1) - 45 = -1 - 5 - 9 - 45 = -60 \][/tex]

Testing [tex]\( x = 3 \)[/tex]:

[tex]\[ f(3) = (3)^3 - 5(3)^2 + 9(3) - 45 = 27 - 45 + 27 - 45 = -36 \][/tex]

Testing [tex]\( x = -3 \)[/tex]:

[tex]\[ f(-3) = (-3)^3 - 5(-3)^2 + 9(-3) - 45 = -27 - 45 - 27 - 45 = -144 \][/tex]

Testing ( x = 5 ):

[tex]\[ f(5) = (5)^3 - 5(5)^2 + 9(5) - 45 = 125 - 125 + 45 - 45 = 0 \][/tex]

[tex]\[ \Rightarrow \text{Root: } x = 5 \][/tex]

Testing ( x = -5 ):

[tex]\[ f(-5) = (-5)^3 - 5(-5)^2 + 9(-5) - 45 = -125 - 125 - 45 - 45 = -340 \][/tex]

Testing ( x = 9 ):

[tex]\[ f(9) = (9)^3 - 5(9)^2 + 9(9) - 45 = 729 - 405 + 81 - 45 = 360 \][/tex]

Testing ( x = -9 ):

[tex]\[ f(-9) = (-9)^3 - 5(-9)^2 + 9(-9) - 45 = -729 - 405 - 81 - 45 = -1260 \][/tex]

Testing ( x = 15 ):

[tex]\[ f(15) = (15)^3 - 5(15)^2 + 9(15) - 45 = 3375 - 1125 + 135 - 45 = 2340 \][/tex]

Testing ( x = -15 ):

[tex]\[ f(-15) = (-15)^3 - 5(-15)^2 + 9(-15) - 45 = -3375 - 1125 - 135 - 45 = -4680 \][/tex]

Testing ( x = 45 ):

[tex]\[ f(45) = (45)^3 - 5(45)^2 + 9(45) - 45 = 91125 - 10125 + 405 - 45 = 81060 \][/tex]

Testing ( x = -45 ):

[tex]\[ f(-45) = (-45)^3 - 5(-45)^2 + 9(-45) - 45 = -91125 - 10125 - 405 - 45 = -102705 \][/tex]

From these tests, we see that ( x = 5 ) is a root of the polynomial. The other roots are irrational or complex.

So, the number of real roots is 1, and it is ( x = 5 ).