To find the amount of salt when the tank contains exactly 40 gallons, create and solve differential equations for salt concentration and tank size over time. We find the tank size is 40 at 10 minutes, at which point there is approximately 14.2 lbs of salt.
Explanation:To solve this, you need to understand that the total amount of salt at any time t is equal to the amount of salt coming in minus the amount of salt going out.
To begin with, the tank has 80 gal x 1/8 lb/gal = 10 lbs of salt.
The amount of salt coming in is 4 gal/min * 1 lb/gal = 4 lbs/min
The amount of salt going out depends on the concentration of the salt in the tank at that time. This is (4-8)(total salt/liters in tank at time t).
Setting up a differential equation and solving gives us an equation for salt concentration and volume at time t:
The equation for the tank size(in gallons) at time t (in minutes) is: tank size = 80 - 4t
The equation for the salt in tank at time t (in minutes) is: salt = 10 - 4t + 80e^-2t
When the tank size is exactly 40 gallons, tank size = 40 = 80 - 4t so t = 10 minutes
Plugging t = 10 into our equation for salt gives us: salt = 10 - 4*10 + 80e^-20 = approximately 14.2 lbs.
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What is the GCF of 12 and 20
Answer:
It is 4.
Step-by-step explanation:
It said so in my classes.
a patio is in the shape of a regular octagon. the sides have length 5m. Calculate the area of the patio
here is the solution.
Round the answer as needed.
Suppose we are flipping a fair coin (i.e., probability of heads = 0.5 and probability of tails = 0.5). further, suppose we consider the result of heads to be a success. what is the standard deviation of the binomial distribution if we flip the coin 5 times?
The nutritional chart on the side of a box of a cereal states that there are 93 calories in a three fourths 3/4 cup serving. How many calories are in 5 cups of the cereal?
The lengths of the sides of a triangle are 4,5,6 can the triangle still be a right triangle
No. We can use the pythagorean theorem to prove this.The equation being a2 + b2 = c2. We then use two of the least of the three numbers, which are 4 and 5, to substitute for “a” and “b”. We get a value for “c” which is 6.4, rounded off to the nearest tenth. This value is greater than 6. Note that this is a very logical way of solving the problem since a greater number for “a” and “b” would lead a greater value for “c”
Write the fractions in order from smallest to largest. 7/10, 3/20,22/25, 2/25
Find all numbers for which the rational expression is undefined z^2+6/z^2-7z-8
Find the volume of the ellipsoid x^2+y^2+9z^2=64
Final answer:
The volume of the ellipsoid x²+y²+9z²=64 is 1024π/3 cubic units.
Explanation:
To find the volume of the ellipsoid given by the equation x²+y²+9z²=64, we can use the formula for the volume of an ellipsoid, which is V = (4/3)πa×b×c, where a, b, and c are the semi-axes of the ellipsoid. In our case, the ellipsoid can be rewritten to show its semi-axes clearly by dividing the equation by 64, which gives us (x²/64) + (y²/64) + (z²/(64/9)) = 1, indicating that our semi-axes are a = b = 8 and c = 8/3. Plugging these values into our volume formula gives us V = (4/3)π×8×8×(8/3), which simplifies to V = 1024π/3 cubic units.
Divide 2x^3 - 35x-12/ x+4
A building has an entry the shape of a parabolic arch 84 ft high and 42 ft wide at the base, as shown below. A parabola opening down with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 84 feet and its width from left to right is 42 feet. Find an equation for the parabola if the vertex is put at the origin of the coordinate system. (1 point)
The equation of the parabola with the given dimensions and vertex at the origin is y = -4x^2.
Finding the Equation of a Parabola
To find the equation of a parabola with a vertex at the origin and given dimensions, we can use the standard form of a parabolic equation, which is y = ax^2. In this case, since the parabola opens downward and the vertex is at the origin (0,0), the equation will have the form y = -ax^2. The value of 'a' can be determined using the dimensions provided for the parabola, which are a width of 42 feet (meaning that the points (21,0) and (-21,0) are on the parabola) and a height of 84 feet (the y-coordinate at the vertex).
Since the point (21, 0) lies on the parabola, substituting it into the equation y=-ax^2 gives us 0 = -a(21)^2, which leads us to find that a = -84/(21)^2. Substituting the value of 'a' back into the equation gives us the final equation of the parabola: y = -84/(21)^2
x^2.
The equation of the parabola is: [tex]\[ y = -\frac{2}{21}x^2 + 42 \][/tex] .
To find the equation of the parabola with its vertex at the origin, we can use the standard form of a parabolic equation, which is [tex]\( y = ax^2 \).[/tex]
Given that the parabola opens downwards, we know that a must be negative. To determine the value of ( a ), we need to find a point on the parabola.
We're given the dimensions of the arch: 84 feet high and 42 feet wide at the base. Since the arch is symmetrical, the highest point is at the midpoint of the base, which is ( x = 0 ). At this point,( y = 84 ).
So, substituting the coordinates of this point into the equation, we get:
[tex]\[ 84 = a \times 0^2 \][/tex]
This simplifies to ( 84 = 0 ), which doesn't give us any useful information. Instead, we need to consider another point on the parabola.
Since the arch is symmetric, we can choose a point where ( x = 21 ) (half of the width of the base), and ( y = 0 ).
Substituting these coordinates into the equation, we get:
[tex]\[ 0 = a \times (21)^2 \][/tex]
0 = 441a
Dividing both sides by 441, we find ( a = 0 ). However, this seems incorrect, as it would mean the arch is just a straight line, which it isn't. This suggests that our choice of coordinates may not be correct.
Let's reconsider. The midpoint of the base is x = 0, but the highest point might not be there. Instead, let's choose a point where ( x = 0 ) and ( y = 42 ), as this is the highest point of the arch.
Substituting these coordinates into the equation, we get:
[tex]\[ 42 = a \times 0^2 \][/tex]
42 = 0
This also doesn't give us useful information. It seems we might have approached this problem incorrectly. Let's try a different strategy.
Since we know the arch is a parabolic shape, and the parabola opens downwards, we can write its equation in the form:
[tex]\[ y = ax^2 + c \][/tex]
To find the values of a and c , we need two points on the parabola. We already have one: the highest point of the arch, which is at x = 0 and (y = 42 ).
Now, we need to find another point. Since the arch is symmetric, we can use any point along the base. Let's choose the point where x = 21 , which is half of the width of the base. At this point, y = 0 .
Substituting these points into the equation, we get:
[tex]\[ 42 = a \times 0^2 + c \][/tex]
[tex]\[ 0 = a \times 21^2 + c \][/tex]
The first equation simplifies to ( c = 42 ).
Substituting this value of ( c ) into the second equation, we get:
[tex]\[ 0 = a \times 21^2 + 42 \][/tex]
Solving for a :
[tex]\[ a \times 441 = -42 \][/tex]
[tex]\[ a = \frac{-42}{441} \][/tex]
[tex]\[ a = -\frac{2}{21} \][/tex]
So, the equation of the parabola is:
[tex]\[ y = -\frac{2}{21}x^2 + 42 \][/tex]
A container holds 5 quarts of lemonade. How much is this in cups?
Paige pays $532 per month for 5 years for a car. she made a down payment of $3,700.00. if the loan costs 7.1% per year compounded monthly, what was the cash price of the car?
Find the quotient. 22x 2 y 2 ÷ 11x 2 y 2
Answer:
Quotient of 22x²y²÷11x²y² is:
2
Step-by-step explanation:
If a number p is divided by b then,
if p could be written as p=qb+r then q is the quotient and r is the remainder
Here, p=22x²y²
and b=11x²y²
22x²y²= 2(11x²y²)+0
⇒ q=2 and r=0
⇒ quotient=2
Hence, quotient of 22x²y²÷11x²y² is:
2
0.8 is 10 times as great as which decimal
How do you write an equation that shows an estimate of each answer for 503+69
To estimate the sum of 503 and 69, one could round the numbers to the nearest tens or hundreds and then add the rounded numbers together. For example, rounding to the nearest tens would result in the equation 500 + 70 = 570.
Explanation:The question asks for a method to write an equation that would allow them to estimate the sum of 503 and 69. This is more related to the concept of rounding numbers. We can estimate this sum by rounding these numbers to the nearest tens or hundreds, and then adding those rounded numbers together.
For example, if you round to the nearest tens, 503 can be rounded down to 500 and 69 can be rounded up to 70. Adding these rounded numbers together gives 500 + 70 = 570. Therefore, one possible equation could be: 500 + 70 = 570, which is a fairly close estimate of the original sum, 503 + 69.
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The equation below represents Function A and the graph represents Function B:
Function A
f(x) = x - 9
Function B
graph of line going through ordered pairs negative 1, negative 3 and 2, 3
Which equation best compares the slopes of the two functions?
Slope of Function B = 2 x Slope of Function A.
Slope of Function A = Slope of Function B
Slope of Function A = 2 x Slope of Function B
Slope of Function B = - Slope of Function A
Answer: The correct option is A., i.e, Slope of Function B = 2 x Slope of Function A.
Explanation:
The given function is,
[tex]f(x)=x-9[/tex]
It can be written as,
[tex]y=x-9[/tex]
It is the slope intercept form like y=mx+c, where m is the slope. On comparing the f(x) with the slope intercept form, we get the slope of f(x) is 1.
The graph of function g(x) passing through the point (-1,-3) and (2,3).
If a line passing through the points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the slope of line is,
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{3-(-3)}{2-(-1)}=\frac{3+3}{2+1}= \frac{6}{3} =2[/tex]
The slope of g(x) is 2.
Since slope of f(x) is 1 and the slope of g(x) is 2, so we can say that the slope of function B is twice of slope of function A.
Slope of Function B = 2 x Slope of Function A
Therefore, the first option is correct.
The answer is A... yeet
A roof rises 9 feet for every 12 feet of run. What is the slope of the roof?
Since slope, m, is defined as the rise/run; then m = 9/12
simplify m = 3/4
In one day,Annie traveled 5 times the sum of the number of hours Brian traveled and 2. Together they traveled 20 hours.Find the number of hours each person traveled.
Would you rather have 1/5 share of £12000
Answer:
Yes bc I want you to have some money
Step-by-step explanation:
A local hamburger shop sold a combined total of 436 hamburger and cheeseburgers on Friday. There were 64 fewer cheeseburgers sold than hamburgers. How many hamburgers were sold on Friday?
64 fewer cheeseburgers were sold than hamburgers. The total number of hamburgers sold on Friday was 250.
Use the concept of subtraction defined as:
Subtraction in mathematics means that is taking something away from a group or number of objects. When you subtract, what is left in the group becomes less.
Given that,
Total number of hamburgers and cheeseburgers sold on Friday: 436.
The number of cheeseburgers sold was 64 fewer than the number of hamburgers.
The objective is to find the number of hamburgers sold on Friday.
To find out how many hamburgers were sold on Friday,
Set up a system of equations.
Let's denote the number of hamburgers as 'H' and the number of cheeseburgers as 'C'.
From the given information,
The total number of hamburgers and cheeseburgers sold is 436,
So write the equation,
H + C = 436.
Since there were 64 fewer cheeseburgers sold than hamburgers,
Which can be written as C = H - 64.
Now substitute the second equation into the first equation and solve for H:
H + (H - 64) = 436
Combine like terms,
2H - 64 = 436
Add 64 to both sides,
2H = 500.
Divide both sides by 2,
H = 250.
Hence,
250 hamburgers were sold on Friday.
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What is the next answer 3,6,4,8,6,12,10
Solve p=10a+3b for a.
What is the area of the trapezoid?
30 square units
60 square units
90 square units
120 square units
Answer:
B-60
Step-by-step explanation:
The area of the trapezoid is 60 square units
How to find the area of a trapezoidThe formula for finding the area of a trapezoid is expressed as:
A = 0.5(a+b)h
Given the following parameters
a = AD = √4² + 6²
AD = √16+36
AD = √52
For the length of BC
a = AD = √6² +10²
AD = √36+100
AD = √136
Calculate the height of the trapezium (10, 6) and (8, 12)
H = √6² +2²
H = √40
Substitute
A = 1/2(7.21+11.66)*6.33
A = 60 square units
Hence the area of the trapezoid is 60 square units
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Describe the two possible ways to use the angle of depression that is outside a right triangle?
Find an equation of the circle whose diameter has endpoints ( −1,−4) and (3,2) .
Final answer:
The equation of the circle with diameter endpoints (-1,-4) and (3,2) is (x - 1)^2 + (y + 1)^2 = 13, found by calculating the center at (1, -1) and the radius as sqrt(52)/2.
Explanation:
To find an equation of the circle whose diameter has endpoints at (-1,-4) and (3,2), we need to determine the center and radius of the circle. The center of a circle is the midpoint of the diameter, and the radius is half the length of the diameter.
First, we calculate the midpoint (which will be the center of the circle) using the formula: (x1 + x2)/2, (y1 + y2)/2. For the given points (-1,-4) and (3,2), the midpoint is ((-1 + 3)/2, (-4 + 2)/2), which simplifies to (1, -1). So, the center of the circle is (1, -1).
To find the radius, we use the distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2). The distance between the two points is sqrt((3 - (-1))^2 + (2 - (-4))^2), which simplifies to sqrt(4^2 + 6^2) = sqrt(16 + 36) = sqrt(52). The radius is half of the diameter, so r = sqrt(52)/2.
The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. Substituting the values we found, we get (x - 1)^2 + (y + 1)^2 = (sqrt(52)/2)^2, which simplifies to (x - 1)^2 + (y + 1)^2 = 13. This is the equation of the circle.
Given a mean of 8 and a standard deviation of 0.7, what is the z-score of the value 9 rounded to the nearest tenth?
Answer: The z-score of the value 9 rounded to the nearest tenth = 1.4
Step-by-step explanation:
Given: Mean [tex]\mu=8[/tex]
Standard deviation [tex]\sigma=0.7[/tex]
The given random value x= 9
Now, the formula to calculate the z score is given by:-
[tex]z=\dfrac{x-\mu}{\sigma}\\\\\Rightarrow\ z=\dfrac{9-8}{0.7}\\\\\Rightarrow\ z=1.42857142857\approx1.4[/tex]
Hence, the z-score of the value 9 rounded to the nearest tenth = 1.4
The physician ordered Polymox (amoxicillin) oral suspension 20 mg/kg/day in divided doses every 8 hours for a child who weights 66 pounds. Convert pounds to kilograms and then calculate the prescribed dosage. ______ mg every _____ hours
How do you write 2/100000 as a decimal?
The fraction 2/100000 written in form of a decimal is 0.00002
Given the fraction 2/100000, we are to express this fraction as a decimal
The fraction can also be expressed as;
2/100000 = 2 * 1/00000
2/100000 = 2 * 0.00001
2/100000 = 0.00002
Hence the fraction 2/100000 written in form of a decimal is 0.00002
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A random sample of 12 graduates of a certain secretarial school typed an average of 79.3 words per minute with a standard deviation of 7.8 words per minute. assuming a normal distribution for the number of words typed per minute, find a 95% confidence interval for the average number of words typed by all graduates of this school.
To find the 95% confidence interval for the average number of words typed, use the formula: sample mean ± (critical value) * (standard deviation / square root of sample size).
Explanation:To find the 95% confidence interval for the average number of words typed by all graduates of the secretarial school, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / square root of sample size)
Plugging in the given values, we have:
Confidence Interval = 79.3 ± (1.96) * (7.8 / √12)
Simplifying, the 95% confidence interval is approximately 75.6 to 83.0 words per minute.
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From home, Mary’s work is two thirds along the way to training. Training is 2.5km from work. Mary normally goes to work, then training and then home again. However, today she forgot her shoes. How far will Mary travel in total today if she has to go home before training to get her shoes?
The total distance Mary would have to travel considering she needs to return home for her shoes is 25km.
Explanation:This math problem involves the calculation of distances within a commute. Here's how we can provide a solution:
We first need to calculate how far Mary's home is to her work. If training is 2.5km from work and this represents one third of the travel (since work is two thirds along the way to training), the total distance from Mary's home to training is 2.5km * 3 = 7.5km. Thus, the distance from Mary's home to work is 7.5km * 2/3 = 5km.So, for Mary's normal route - home to work, then work to training, then training to home - she travels 5km (home to work) + 2.5km (work to training) + 7.5km (training to home) = 15km.Today, however, Mary needs to go back home from work for her shoes before she can proceed to training. This path would be: home to work (5km), work to home (5km), home to training (7.5km), and finally, training to home (7.5km) giving a total of 5km + 5km + 7.5km + 7.5km = 25km. So, Mary will travel a total of 25km today due to the forgotten shoes.Learn more about Distance calculation here:
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