Answer:
The quadratic formula is derived from a quadratic equation in standard form when solving for x by completing the square. The steps involve creating a perfect square trinomial, isolating the trinomial, and taking the square root of both sides. The variable is then isolated to give the solutions to the equation.
Step-by-step explanation:
Jen bought 5 cds for 57.50. What was the unit price
The sum of -4 and 8 times a number is multiplied by 3. The result is 15 less that 25 times the number. What is the number?
A) -12
B) 6/7
C) 0
D) 3
E) none of the above
The population of a community is known to increase at a rate proportional to the number of people present at time t. if an initial population p0 has doubled in 7 years, how long will it take to triple? (round your answer to one decimal place.)
To determine how long it will take for the population to triple, we need to solve the equation 3P0 = P0e^(rt) for t. Rounding the answer to one decimal place, it will take approximately 10.1 years for the population to triple.
Explanation:To solve this problem, we can use the equation P = P0e^(rt), where P is the population at a given time, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), r is the growth rate, and t is the time in years.
In this case, the population has doubled in 7 years, so we can write the equation as 2P0 = P0e^(r*7). We need to solve for r.
Dividing both sides of the equation by P0 gives us 2 = e^(7r). Taking the natural logarithm of both sides gives us ln(2) = 7r. Solving for r, we find that r ≈ ln(2)/7.
Now, to determine how long it will take for the population to triple, we need to solve the equation 3P0 = P0e^(r*t) for t. Substituting the value of r we found, we get 3 = e^((ln(2)/7)*t). Taking the natural logarithm of both sides to solve for t, we have ln(3) = (ln(2)/7)*t. Solving for t, we find that t ≈ 7 * ln(3)/ln(2). Rounding the answer to one decimal place, it will take approximately 10.1 years for the population to triple.
It takes approximately 11.1 years for the population to triple.
The population growth can be modeled by the exponential function [tex]P(t)=P_0 \times e^{k t}[/tex], where [tex]P_0[/tex] is the initial population, k is the growth rate, and t is time.
In this case, since the population doubles in 7 years, we have the relationship P(7) = 2 x [tex]P_0[/tex].
[tex]P(7)=P_0 \times e^{7 k}[/tex]
[tex]2 \times P_0=P_0 \times e^{7 k}[/tex]
Divide both sides by [tex]P_0[/tex].
[tex]2=e^{7 k}[/tex]
Take the natural logarithm (ln) of both sides to solve for k:
ln(2) = 7k
[tex]k=\frac{\ln (2)}{7}[/tex]
Now that we have the growth rate k, we can use it to find the time it takes for the population to triple.
[tex]P(t)=P_0 \times e^{k t}[/tex]
For tripling, [tex]P(t)=3 \times P_0[/tex].
[tex]3 \times P_0=P_0 \times e^{k t}[/tex]
[tex]3=e^{k t}[/tex]
Take the natural logarithm again:
ln(3) = kt
[tex]t=\frac{\ln (3)}{k}[/tex]
Substitute the value of k we found earlier:
[tex]t=\frac{\ln (3)}{\frac{\ln (2)}{7}}[/tex]
Using natural logarithm values:
[tex]t \approx \frac{1.0986}{\frac{0.6931}{7}}[/tex]
[tex]t \approx \frac{1.0986}{0.0990}[/tex]
t ≈ 11.09
24% of 1321 is how much? round the answer to the nearest hundredth
If the federal reserve decreases the reserve rate from 7% to 5%, how does this affect the amount of money that would result because of fraction-reserve rbanking from an initial deposit into a bank of $30,000
When the Federal Reserve decreases the reserve rate from 7% to 5%, the amount of money resulting from fraction-reserve banking increases. In this case, an initial deposit of $30,000 would result in $28,500 due to the change in reserve rate.
Explanation:When the Federal Reserve decreases the reserve rate from 7% to 5%, it means that banks are required to hold a smaller percentage of deposits as reserves. This change allows banks to lend out a larger portion of the initial deposit. In this case, if the initial deposit into a bank is $30,000, the amount that would result from fraction-reserve banking would increase because the bank can now lend out a greater percentage of the deposit.
To calculate the new amount that would result from fraction-reserve banking, we need to calculate the reserve amount and the loan amount.
Therefore, the amount that would result from fraction-reserve banking from an initial deposit of $30,000 would be $28,500.
Learn more about Federal Reserve reserve rate and fraction-reserve banking here:https://brainly.com/question/30984737
#SPJ2
Six identically shaped place mats, marked a, b, c, d, e and f, are piled neatly in a stack. how many arrangements are possible if a and b must touch each other?
90 points correct answer
If the parent function f(x) = x2 is modified to g(x) = 2x2 + 1, which statement is true about g(x)?
It is an even function.
It is an odd function.
It is both an even and an odd function.
It is neither an even nor an odd function.
Tom and Rob are brothers who like to make bets about the outcomes of different contests between them. Before the last bet, the ratio of the amount of toms money to the amount of robs money was 4:7. Rob lost the latest competition, and now the ratio of the amount of robs money is 8:3. If rob had $280 before the last competition, how much rob have now that he lost the bet?
before the bet Rob had 7 out of 11 ( 4+7)
280/7 = 40
after bet Rob had 3
3*40 = 120
he has $120
let f(x)=4x-7 and g(x)-5+9x. what are f+g and f-g? what are their domains?
Round 156, 782 to nearest thousand and the nearest ten thousand.
The answer is 167,000 because if you round to the 6,782 up it is 7,000. If you round 57,782 up it is 60,000. so then just add those 2 together and plug the 100,000 into it.
Kwamee bought kona beans that cost 11.00 and bought fuji beans that cost 7.50.he bought a total of 23 pounds of coffee and it cost 197.00 write a system of equations to solve how many pounds of kona and fuji beans kwamee bought
The equation 14m = 400.40 can be used to find the cost per yard m, in dollars, of mulch. what is the value of m? m = $
Prove that composition of injective functions is injective. prove that composition of surjective functions is surjective.
It is found that the composition of injective functions is injective.
What is an injective function?An injective function is not a surjective function. This means that we want a function that has a unique output for each input, that doesn't cover the natural numbers.
Proof 1: Proving the composition of injective functions is injective.
f(x) is injective
f(a) = f(b)
a = b
Thus g(x) is injective
g(a) = g(b)
a = b
To prove; f(g(x)) is injective
f(g(a)) = f(g(b))
f(a) = f(b)
Since, g(x) is injective
a = b
Therefore, if f(g(a)) = f(g(b)), then a = b
That proving g(f(x)) is injective is going to follow a similar path of steps.
Proof 2: Proving the composition of surjective functions is surjective.
f(x) is surjective
The range of the function of x is the set of all real numbers
g(x) is surjective
The range of the function of x is the set of all real numbers
f(g(x)) is also surjective. The range of f(x) is the set of all real numbers. where g(x) represents all reals as well means that we touch all of the inputs needed to generate all the outputs for f(x).
Proving g(f(x)) to be surjective follows a similar path.
Learn more about function here:
https://brainly.com/question/2253924
#SPJ5
what hourly rate does the repair shop charge for labor?
The mechanic's hourly rate is $55.
Understanding Labor Costs in a Repair Shop
To determine the hourly rate a repair shop charges for labor, we must analyze the information given and apply simple arithmetic operations. According to the examples provided, an auto mechanic charges a flat fee and an hourly rate for labor. For instance, a mechanic charging a flat fee of $75 per car, plus an hourly rate of $55 can be represented by the linear equation y = 75 + 55x, where 'y' is the total labor charge, and 'x' is the number of hours worked.
Example Calculation
Using the equation above, if the job takes 3.5 hours, the labor charge (y) would be calculated as $75 + ($55 imes 3.5) = $75 + $192.50 = $267.50. Hence, we can infer that the mechanic's hourly rate is $55.
If a(x)=3x+1 and b(x)=
The answer is C. [1, ∞] on edge
How to solve equations
How to find mass when given density and volume?
6=x+2/3
Please help ASAP!!! 10 Points!!! :)
A line passes through the point (–2, 7) and has a slope of –5. What is the value of a if the point (a, 2) is also on the line?
A.–7
B.–1
C.1
D.7
Answer:
The correct option is B.
Step-by-step explanation:
The slope of a line is constant.
If a line passes through two points then the slope of the line is
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
It is given that the line passes through (-2,7) and (a,2).
Using the above formula the slope of the line is
[tex]m=\frac{2-7}{a-(-2)}[/tex]
[tex]m=\frac{-5}{a+2}[/tex]
It is given that the slope of the line is -5. Substitute m=-5 in the above equation.
[tex]-5=\frac{-5}{a+2}[/tex]
On cross multiplication we get
[tex]-5(a+2)=-5[/tex]
Divide both sides by -5.
[tex]a+2=1[/tex]
Subtract 2 from both the sides.
[tex]a=1-2[/tex]
[tex]a=-1[/tex]
The value of a is -1. Therefore the correct option is B.
Answer:
The answer is B.
Step-by-step explanation:
Which graph represents the system of inequalities?
The product of (a − b)(a − b) is a2 − b2.
Sometimes
Always
Never
Final answer:
The statement that the product of (a − b)(a − b) equals [tex]a^2 - b^2[/tex] is never true; the correct product is always [tex]a^2 - 2ab + b^2[/tex], representing the square of a difference formula.
Explanation:
The statement "The product of (a − b)(a − b) is [tex]a^2 - b^2[/tex]." is never true. Instead, using the distributive property to expand the expression (a − b)(a − b) yields a different result, as each term in the first binomial must be multiplied by each term in the second binomial.
The correct expansion is:
(a − b)(a − b) = a(a − b) - b(a − b)= aa - ab - ba + bb= [tex]a^2 - 2ab + b^2[/tex](since ab and ba are equivalent)Thus, the product of (a − b)(a − b) is always [tex]a^2 - 2ab + b^2[/tex], which is the square of a difference formula, not [tex]a^2 - b^2[/tex] which is the difference of squares formula. The latter is applicable to the product (a + b)(a - b).
Which equation is equivalent to
X/ +11 = 15
If f(x)=3/x+2-root x-3 complete the following statement (round answer to the nearest hundredth) f(7)=
Find the value of x in the equation 2√ x
a. –5
b. 1
c. 16
d. 25 11. simplify the expression .
a.
b.
c.
d. + 20 = 15 + 3√ x
Summer paints a room that has 400 square feet of wall space in 2.5 hours. at this rate, how long will it take her to paint a room with 720 square feet of wall space?
Find the quotient.
110/10=
Identify where the given information is sufficient for graphing a linear equation. Check all that apply.
A.one point and the slope of the line
B.one point
C.one of the intercepts and the slope of the line
D.both the intercepts
E.two points
To graph a linear equation, you need at least two pieces of information: the slope and a point, both intercepts, or two points. One point and the slope, or one intercept and the slope, are also sufficient to graph a linear equation.
Explanation:For graphing a linear equation, there are several options that provide sufficient information:
A. One point and the slope of the lineC. One of the intercepts and the slope of the lineD. Both the interceptsE. Two pointsTo graph a linear equation, you need at least two pieces of information, such as the slope and a point or both intercepts. Having two points is also sufficient since it helps determine the slope. Having just one point or one intercept along with the slope is also enough to graph the line.
A quota is when one country
Could -2 be used as a base to rewrite 32? 64? Why or why not?
Is g(x)= 5 + 6/x a linear function