Step-by-step explanation:
The point-slope form of an equation of a line:
[tex]y-y_1=m(x-x_1)[/tex]
m - slope
(x₁, y₁) - point on a line
We have the point (3, -3) and the slope m = -2. Substitute:
[tex]y-(-3)=-2(x-3)[/tex]
[tex]y+3=-2(x-3)[/tex] - point-slope form
Convert to the slope-intercept form (y = mx + b):
[tex]y+3=-2(x-3)[/tex] use the distributtive property
[tex]y+3=-2x+(-2)(-3)[/tex]
[tex]y+3=-2x+6[/tex] subtract 3 from both sides
[tex]y=-2x+3[/tex] - slope-intercept form
Convert to the standard form (Ax + By = C):
[tex]y=-2x+3[/tex] add 2x to both sides
[tex]2x+y=3[/tex] - standard form
Convert to the general form (Ax + By + C = 0):
[tex]2x+y=3[/tex] subtract 3 from both sides
[tex]2x+y-3=0[/tex] - general form
7. Mis the midpoint of QR and M has
coordinates (-2, 6). Q has coordinates
(8, -10). What are the coordinates of R?
nges to the original content are the responsibility of the instructor.
I need help on number 7 please
Answer:
R(- 12, 22 )
Step-by-step explanation:
Using the midpoint formula
[tex]\frac{1}{2}[/tex](8 + [tex]x_{R}[/tex] ) = [tex]x_{M}[/tex] = - 2
Multiply both sides by 2
8 + [tex]x_{R}[/tex] = - 4 ( subtract 8 from both sides )
[tex]x_{R}[/tex] = - 12
----------------------------------------------
[tex]\frac{1}{2}[/tex](- 10 + [tex]y_{R}[/tex] ) = [tex]y_{M}[/tex] = 6
Multiply both sides by 2
- 10 + [tex]y_{R}[/tex] = 12 ( add 10 to both sides )
[tex]y_{R}[/tex] = 22
The coordinates of R = (- 12, 22 )
Which shows a perfect square trinomial?
502-4x2
100-36x?y
16x2+24 xy +9y2
49x2 - 70 xy +10y
Answer:
Third choice.
Step-by-step explanation:
Trinomial means you have 3 terms.
You don't have 3 terms in first two choices so let's not look at them.
Anything of the form [tex]a^2x^2+2abxy+ b^2y^2[/tex] is a perfect square trinomial because it can be written as [tex](ax+by)^2[/tex].
Let's this this:
[tex](ax+by)^2[/tex]
[tex](ax+by)(ax+by)[/tex]
Now foil!
First=ax(ax)=a^2x^2
Outer=ax*by=abxy
Inner=by*ax=abxy
Last=by*by=b^2y^2
Add together and this gives you a^2x^2+2abxy+b^2y^2.
So looking at third choice you can write it as 4^2x^2+24xy+3^2y^2.
Is 2*4*3 equal to 24? If it is then you have your answer. It is.
We have our answer.
the rational roots of a polynomial function f(x) can be written in the form p/q where p is a factor of the leading corfficient of the polynomial and q is a factor of the constant term true or false
False.
The Rational Root theorem states that P is a factor of the constant term and q is a factor of the leading coefficient.
What is the average rate of change for the sequence shown below?
coordinate plane showing the points 1, 4; 2, 2.5; 3, 1; and 4, negative 0.5
Answer:
[tex]\large\boxed{-1\dfrac{1}{2}}[/tex]
Step-by-step explanation:
The points on the graph are collinear (they lie on one straight line).
Therefore, average of change is the same as a slope.
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
We choose two points and put the coordinates to the formula
[tex](1, 4), (3, 1)\\\\\dfrac{1-4}{3-1}=\dfrac{-3}{2}=-1\dfrac{1}{2}[/tex]
The average rate of change for the sequence given is [tex]-1\frac{1}{2}[/tex].
What is the average rate of change?The average rate of change formula is used to find the slope of a graphed function. To find the average rate of change, divide the change in y-values by the change in x-values.
The points on the graph are collinear (they lie on one straight line).
Therefore, average of change is the same as a slope.
The formula of a slope:
[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
We choose two points and put the coordinates to the formula
(1, 4), (3, 1)
[tex]\frac{1-4}{3-1} =\frac{-3}{2} =-1\frac{1}{2}[/tex]
The average rate of change for the sequence given is [tex]-1\frac{1}{2}[/tex].
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Which polynomial is prime?
X2-36
X2-16
X2-7x + 12
X2-X-20
Answer:
Step-by-step explanation:
x^2 - 36 is the difference of two squares and factors as follows:
(x - 6)(x + 6)
x^2 - 16 is the difference of two squares and factors as follows:
(x - 4)(x + 4)
x^2 - 7x + 12 is an easily factored quadratic; the factors are
(x - 3)(x - 4)
x^2 - x - 20 is an easily factored quadratic; the factors are
(x - 5)(x + 4)
I conclude that none of the four expressions is prime.
Answer:
B. [tex]x^2+16[/tex]
Step-by-step explanation:
We are asked to find the prime polynomial from our given choices.
We know that a polynomial is prime, when it has only two factors that are 1 and polynomial itself.
Upon looking at our given choices, we can see that each polynomial can be factored except [tex]x^2+16[/tex].
We can see that [tex]x^2+16[/tex] is sum of squares and sum of squares cannot be factored, therefore, polynomial [tex]x^2+16[/tex] is a prime polynomial.
PLS ANSWER THESE QUESTIONS. I WILL GIVE 20 POINTS AD BRAINLIEST.
1. Write an expression to represent the sum of three consecutive even numbers. Let x equal the first number.
2. Find the circumference of a circle that has a diameter of 1 4/5 inches. Use 22/7 for pi.
3. Divide. Write the quotient in simplest form. 1/7 divided by -2/7 = ?
THANK YOU
Answer:
1. 3x + 6
2. [tex]\frac{99}{14}[/tex] or 7.071
3. [tex]-\frac{1}{2}[/tex]
Step-by-step explanation:
1. 3 consecutive even integers
If the first is x, the 2nd is x+2, and the 3rd is x+4
x + (x+2) + (x+4)
= 3x + 6
2. The equation for circumference is
[tex]C=2\pi r[/tex]
If the diameter is [tex]1\frac{4}{5} = \frac{9}{5}[/tex]
The the radius is half of that.
So the circumference is
[tex]\pi * \frac{9}{4} *\frac{1}{2} * 2\\= \pi * \frac{9}{4}\\=\frac{22}{7} * \frac{9}{4}\\= \frac{198}{28}\\ =\frac{99}{14} \\=7.071[/tex]
3. To divide we multiply by the reciprocal. So flip the fraction that we are dividing by.
[tex]\frac{1}{7} / -\frac{2}{7}\\ = \frac{1}{7} * -\frac{7}{2} \\= -\frac{1}{2}[/tex]
Which point is on the graph of f(x) = 3 • 4x? A. (0, 12) B. (0, 0) C. (1, 12) D. (12, 1)
Answer:
(1,12) is correct if you meant [tex]3 \cdot 4^x[/tex].
Please correct if I'm wrong about your expression.
Step-by-step explanation:
I think you mean [tex]f(x)=3 \cdot 4^x[/tex].
Let's test the point and see.
A. (0,12)?
(0,12)=(x,y)
What happens when x equals 0? Is the result 12?
[tex]3 \cdot 4^0[/tex]
[tex]3 \cdot 1[/tex]
[tex]3(1)[/tex]
[tex]3[/tex]
Yep that isn't 12 so (0,12) is not on the graph of f.
B. (0,0)?
(0,0)=(x,y)
What happens when x equals 0? Is the result 0?
[tex]3 \cdot 4^0[/tex]
We already this and got 3 so (0,0) is not on the graph of f.
C. (1,12)?
(1,12)=(1,12)
What happens when x equals 1? Is the result 12?
[tex]3 \cdot 4^1[/tex]
[tex]3 \cdot 4[/tex]
[tex]12[/tex]
The result is 12 so (1,12) is on the graph of f.
C. (12,1)
(12,1)=(x,y)
What happens when x equals 12? Is the result 1?
[tex]3 \cdot 4^{12}[/tex]
This will result in a really big number that isn't 1 so (12,1) is not on the graph of f.
(1,12) is correct if you meant [tex]3 \cdot 4^x[/tex].
Answer:
(1,12)
Step-by-step explanation:
ape x
The sum of two numbers is 10 the larger number is four times the smaller number this is the system of equations used to represent the scenario is Y equals negative X +10 and Y equals 4X what is the larger number
Answer:
The larger number is 8
Step-by-step explanation:
Let
x -----> the smaller number
y ----> the larger number
we know that
x+y=10
y=10-x -----> equation A
y=4x -----> equation B
Solve by substitution
substitute equation B in equation A and solve for x
4x=10-x
4x+x=10
5x=10
x=2
Find the value of y
y=4x-------> y=4(2)=8
therefore
The smaller number is 2 and the larger number is 8
To find the larger number where the sum of two numbers is 10 and the larger is four times the smaller, the system of equations y = -x + 10 and y = 4x is used, and through solving, the larger number is determined to be 8.
The question entails finding the larger number when two numbers sum to 10, and the larger number is four times the smaller number. The system of equations representing the scenario is y = -x + 10 and y = 4x. To find the larger number, set these two equations equal to each other since they both equal y:
4x = -x + 10
Now, solve for x by adding x to both sides:
5x = 10
Then, divide both sides by 5:
x = 2
Since x is the smaller number, and y is four times larger, compute y:
y = 4x = 4(2) = 8
The larger number (y) in this case is 8.
Factorize a^2+b^2+2(ab-ac-bc)
Answer:
(a + b)(a + b - 2c)
Step-by-step explanation:
Note that
(a + b)² = a² + b² + 2ab
Given
a² + b² + 2(ab - ac - bc) ← distribute parenthesis
= a² + b² + 2ab - 2ac - 2bc
= (a + b)² - 2ac - 2bc ← factor out - 2c from each term
= (a + b)² - 2c (a + b) ← factor out (a + b) from each term
= (a + b) [ a + b - 2c ]
= (a + b)(a + b - 2c) ← in factored form
The expression a² + b² + 2(ab - ac - bc) cannot be factorized using standard factorization techniques over real numbers, as a² + b² is not factorizable over the reals and there's no common factor for all terms.
To factorize the expression a² + b²+ 2(ab - ac - bc). To factorize this, we will look for a common factor and regroup the terms. Let's see if there's a way to rearrange the terms to resemble a known pattern or factor by grouping.
First, let's rewrite the expression by grouping terms with a common factor:
a² + 2ab - 2ac
b² - 2bc
However, we notice that the expression does not fit into a perfect square or any other easily factorizable form like (a + b)² or (a - b)² due to the nature of the terms a², b², and 2(ab - ac - bc). The expression is already in its simplest factored form as it stands because a²+ b² is not factorizable over the real numbers, and there's no common factor for all terms.
Therefore, the expression a² + b² + 2(ab - ac - bc) does not factorize further using real numbers and the usual factorization techniques.
Will mark brainliest, please answer:)
Find the value of AG. Round to the nearest tenths if necessary. Explain work.
(Image of Question is above and use the 3D Pythagorean Theorem rule)
Check the picture below.
How many degrees would a square need to be rotated to map onto itself?
Answer:
360
Step-by-step explanation:
what are terms? how do you combine them? can you give me an example?
A term is any number, variable or combination of a number and a variable in an equation.
Examples:
In 2x + 5y + 8, 2x is one term, 5y is a second term and 8 is a 3rd term.
In the equation 2x + 3 + 5x -2
Combine the like terms 2x and 5x are like terms because they both have x as a variable, there is a plus sign in fron of the 5x, so you would have 2x +5x = 7x
Then 3 and 3 are like terms, because they are just numbers, there is a subtraction sign in front of the 2, so you have 3-2 = 1
The equation then becomes: 2x + 3 + 5x -2 : 7x + 1
If y = x+ 5 were changed to y = x + 9, how would the graph of the new
function compare with the first one?
O
A. It would be shifted down.
O
B. It would be shifted up.
O
C. It would be shifted right.
O
D. It would be steeper.
Answer: B. It would be shifted up.
Step-by-step explanation: We are changing the last number only. This number determines where the y-intercept is, which is on the vertical axis. Since the number increases by 4, the y-intercept would be shifted up by 4, without changing the slope. Therefore, the answer would be B. It would be shifted up.
Solve the right triangle, ΔABC, for the missing sides and angle to the nearest tenth given angle B = 39° and side c = 13.
Answer:
Part 1) [tex]b=8.2\ units[/tex]
Part 2) [tex]a=10.1\ units[/tex]
Part 3) [tex]A=51\°[/tex] and [tex]C=90\°[/tex]
Step-by-step explanation:
see the attached figure with letters to better understand the problem
step 1
Find the side b
we know that
In the right triangle ABC
The function sine of angle B is equal to divide the opposite side angle B (AC) by the hypotenuse (AB)
[tex]sin(B)=AC/AB[/tex]
we have
[tex]AB=c=13\ units[/tex]
[tex]AC=b[/tex]
[tex]B=39\°[/tex]
substitute
[tex]sin(39\°)=b/13[/tex]
solve for b
[tex]b=(13)sin(39\°)[/tex]
[tex]b=8.2\ units[/tex]
step 2
Find the side a
we know that
In the right triangle ABC
The function cosine of angle B is equal to divide the adjacent side angle B (BC) by the hypotenuse (AB)
[tex]cos(B)=BC/AB[/tex]
we have
[tex]AB=c=13\ units[/tex]
[tex]BC=a[/tex]
[tex]B=39\°[/tex]
substitute
[tex]cos(39\°)=a/13[/tex]
solve for a
[tex]a=(13)cos(39\°)[/tex]
[tex]a=10.1\ units[/tex]
step 3
Find the measure of angle A
we know that
In the right triangle ABC
[tex]C=90\°[/tex] ----> is a right angle
[tex]B=39\°[/tex]
∠A+∠B=90° ------> by complementary angles
substitute the given value
[tex]A+39\°=90\°[/tex]
[tex]A=90\°-39\°[/tex]
[tex]A=51\°[/tex]
please
simplify 7^6 √ 7^2
Answer:
Step-by-step explanation:
The tough part of this question is figuring out what to do with √(7^2). You could do it by expanding the square. √(7^2) = √49
Now what is the square root of 49? Is it not 7?
√49 = 7
7^6 * 7^1
7 ^(6 + 1)
7 ^ 7
Answer:
49
Step-by-step explanation:
Plutonium-240 decays according to the function L where o
represents the quantity remaining after t years and k is the decay constant,
0.00011... How long will it take 24 grams of plutonium-240 to decay to 20
grams?
Answer:
[tex]\boxed{\textbf{1700 yr}}[/tex]
Step-by-step explanation:
[tex]-\dfrac{\text{d}L}{\text{d}t} = kL\\\\\dfrac{dL}{L}=-kdt\\\\\ln L = -kt + C\\\text{At t = 0, L = L$_{0}$, so C = $\ln L_{0}$}\\\ln L = -kt + \ln L_{0}\\\ln L_{0} - \ln L = kt\\\\\ln \dfrac{L_{0}}{L} =kt[/tex]
Data:
L₀ = 24 g
L = 20 g
k = 0.000 11 yr⁻¹
Calculation:
[tex]\ln \dfrac{24}{20} =0.000 11t\\\\\ln 1.2 = 0.000 11t\\\\0.1823 = 0.000 11t\\\\t = \dfrac{0.1823}{0.000 11} = \textbf{1700 yr}\\\\\text{It will take } \boxed{\textbf{1700 yr}}\text{ for the polonium to decay to 20 g}[/tex]
Edgar accumulated $5,000 in credit card debt. If the interest is 20% per year and he does not make any payment for 2 years. How much will he owe on this debt in 2 years by compounding continously?
Answer:
$7,434.57
knewton alta 2023
Step-by-step explanation:
Edgar will owe approximately $6,360.92 on his credit card debt in 2 years with continuous compounding.
Explanation:To calculate the amount Edgar will owe on his credit card debt in 2 years with continuous compounding, we can use the formula for compound interest:
A = P*e^(rt)
Where:
A is the final amountP is the initial principal (the amount Edgar owes)e is Euler's number (approximately equal to 2.71828)r is the interest rate per year in decimal formt is the time in yearsPlugging in the values, we have:
A = $5,000 * e^(0.20*2)
Calculating this expression gives the approximate value of $6,360.92. Therefore, Edgar will owe approximately $6,360.92 on his credit card debt in 2 years with continuous compounding.
Solve y over negative 2 + 5 = 13
Answer: y=-16
Step-by-step explanation:
Y/-2+5=13
Y/-2=8
Y=-16
Terry has 2 more quarters than fines and has a total of $6.80. How many quarters and dimes does Terry have?
Answer:
Terry has 20 quarters and 18 dimes
Step-by-step explanation:
Let
x -----> the number of quarters
y ----> the number of dimes
Remember that
1 quarter=$0.25
1 dime=$0.10
we know that
x=y+2 ----> equation A
0.25x+0.10y=6.80 -----> equation B
Substitute equation A in equation B and solve for y
0.25(y+2)+0.10y=6.80
0.25y+0.50+0.10y=6.80
0.25y+0.10y=6.80-0.50
0.35y=6.30
y=18 dimes
Find the value of x
x=y+2 -----> x=18+2=20 quarters
therefore
Terry has 20 quarters and 18 dimes
To determine the number of quarters and dimes Terry has, set up two equations based on the information provided, solve one of the equations for one variable, and substitute this into the second equation. Solve the equation to get the number of dimes and substitute it into the first equation to get the number of quarters.
Explanation:To solve this, we can use algebra, setting up equations to represent the problem and then solve it.
Let F represent the number of dimes (since a dime is worth $0.10) and let Q represent the number of quarters (since a quarter is worth $0.25). We have two key pieces of information:
Terry has 2 more quarters than dimes: Q = F + 2The total amount of money Terry has equals $6.80: 0.10F + 0.25Q = 6.80From the first equation, we can substitute F + 2 for Q in the second equation: 0.10F + 0.25(F + 2) = 6.80.
Solve this equation to find the value of F, representing the number of dimes, Terry has. Then, substitute the value of F into the first equation to determine the number of quarters Terry has.
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What is the solution to the equation? 305p equals 1,525
Answer:
Step-by-step explanation:
305p=1,525
305 305
p=5
The solution of the linear equation 305p = 1525 will be 5.
What is the linear system?A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.
The linear equation with one variable is given below.
305p = 1525
On simplifying, we have
305p = 1525
p = 1525 / 305
p = 5
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Use the quadratic formula to solve the equation -3x^2-x-3=0
Answer:
[tex]x_{1=} \frac{-1+i\sqrt{35} }{6} \\\\x_{2=} \frac{-1-i\sqrt{35} }{6}[/tex]
Step-by-step explanation:
Using the quadratic formula:
[tex]x=\frac{-b+-\sqrt{b^{2}-4*a*c} }{2*a}[/tex]
We will have two solutions:
[tex]x_{1}\\ x_{2}[/tex]
-3x^2-x-3=0 a=-3 b=-1 c=-3
We have:
[tex]x_{1}=\frac{1+\sqrt{-35} }{-6}\\\\x_{2}=\frac{1-\sqrt{-35} }{-6}\\[/tex]
we can write:
[tex]x_{1}=\frac{-1+\sqrt{-35} }{6}\\\\x_{2}=\frac{-1-\sqrt{-35} }{6}\\[/tex]
The solutions are not real numbers.
So, we know: [tex]i=\sqrt{-1}[/tex]
Finally we have:
[tex]x_{1}=\frac{-1+i\sqrt{35} }{6}\\\\x_{2}=\frac{-1-i\sqrt{35} }{6}\\[/tex]
If a fixed number is added to each term of an arithmetic sequence, is the resulting sequence an arithmetic sequence? explain
Answer:
Yes.
Step-by-step explanation:
If we add a fixed number to each term of an arithmetic sequence, we are still going to be having an arithmetic sequence.
For example, given the following sequence:
1, 3, 5, 7, 9, 11...
The difference between consecutive terms is 2, therefore the pattern is adding two to the previous term.
If we add a fix number, let's say '3':
1+3, 3+3, 5+3, 7+3, 9+3, 11+3...
4, 6, 8, 10, 12, 14...
We notice that the pattern is the same, and it's still an arithmetic sequence.
What is the length of the side of a right triangle that has a side length of 12 ft and hypotenuse that measures 15 ft
Answer:
9 ft
Step-by-step explanation:
The Pythagorean theorem states
a^2+b^2 = c^2 where a and b are the legs and c is the hypotenuse
Substituting in what we know (one leg is 12 and the hypotenuse is 15)
12^2 +b^2 = 15^2
144+ b^2 = 225
Subtract 144 from each side
144-144 +b^2 = 225-144
b^2 =81
Take the square root of each side
sqrt(b^2) = sqrt(81)
b = 9
Find four consecutive even integers with a sum of -52
Final answer:
To find four consecutive even integers with a sum of -52, we denote the smallest integer as x and use the equation x + (x+2) + (x+4) + (x+6) = -52 to find that these integers are -16, -14, -12, and -10.
Explanation:
To find four consecutive even integers with a sum of -52, let's denote the smallest of these integers as x. Consequently, the next three integers can be represented as x+2, x+4, and x+6. The sum of these four integers can be expressed as the equation x + (x+2) + (x+4) + (x+6) = -52.
Combining like terms gives us 4x + 12 = -52. Subtracting 12 from both sides gives 4x = -64, and dividing both sides by 4 yields x = -16. Therefore, the four consecutive even integers are -16, -14, -12, and -10.
The four consecutive even integers with a sum of -52 are -16, -14, -12, and -10.
Let's denote the four consecutive even integers as [tex]\( x \), \( x+2 \), \( x+4 \), and \( x+6 \)[/tex].
According to the problem, their sum is -52. So, we can set up the equation:
[tex]\[ x + (x + 2) + (x + 4) + (x + 6) = -52 \][/tex]
Now, let's solve for \( x \):
[tex]\[ 4x + 12 = -52 \][/tex]
Subtract 12 from both sides:
[tex]\[ 4x = -52 - 12 \][/tex]
[tex]\[ 4x = -64 \][/tex]
Divide both sides by 4:
[tex]\[ x = \frac{-64}{4} \][/tex]
[tex]\[ x = -16 \][/tex]
Now that we've found the value of \( x \), we can find the consecutive even integers:
- First even integer: [tex]\( x = -16 \)[/tex]
- Second even integer: [tex]\( x + 2 = -16 + 2 = -14 \)[/tex]
- Third even integer: [tex]\( x + 4 = -16 + 4 = -12 \)[/tex]
- Fourth even integer: [tex]\( x + 6 = -16 + 6 = -10 \)[/tex]
So, the four consecutive even integers with a sum of -52 are -16, -14, -12, and -10.
complete question given below:
Find four consecutive even integers with a sum of -52.Find the four integeres
Neeeed help noww please
Answer:
6
Step-by-step explanation:
f(n+1) = f(n) -2
We know f(1) = 10
Let n=1
f(1+1) = f(1) -2
f(2) = 10-2 =8
We now know f(2)
Let n=2
f(2+1) = f(2)-2
f(3) = 8 -2=6
f(3) =6
Find the measure of the third angle of a triangle given that the first two angles are 44º and 72º.
Show your work.
The sum of the three angles of a triangle need to equal 180
To find the third angle subtract the two known angles from 180.
Third angle = 180 - 44 - 72 = 64 degrees.
Answer:
All angles of a triangle add up to 180. Just subtract. -72 --> 108 - 44. 64
There are 134third grades and 167 fourth -grades at the annual school and family picnic. The number of students is 7 times the number of adults. Each picnic table can seat 9 people. How many picnic tables will need to be set up for the picnic????
Answer:
39 tables
Step-by-step explanation:
Given:
third grade= 134
fourth grade=167
total=134+167=301
Now no. of students is 7 times no. of adults,
no. of adults= 301/7
=43
Total no. of people =301+43
=344
Each picnic table can seat 9 people, so
No. of tables for 344 people= 344/9
=38.222
39 tables picnic tables will need to be set up for the picnic!
What is the measure of angle ABC in the circle shown in the picture?
Answer:
45°
Step-by-step explanation:
I think you meant m<ACB, and from what I see here, I took half of 90°, which is 45°.
What is the solution to the equation ? 1/2n =8 ? = 4 n = 6 n = 10 n = 16
Answer:
n = 16
Step-by-step explanation:
1/2 n = 8 (multiply both sides by 2)
(1/2) (2) n = 8 (2)
n = 8 (2)
n = 16
The distance traveled, in feet, of a ball dropped from a tall building is modeled by the equation d(t) = 16t2 where d equals the distance traveled at time t seconds and t equals the time in seconds. What does the average rate of change of d(t) from t = 2 to t = 5 represent?\
Answer:
The rate of change determines the average speed of the ball when it is dropped from the building.
Step-by-step explanation:
d(t) = 16t^2
when t = 2
d(t) = 16 (2)²
d(t) = 64
when t = 5
d(t) = 16 (5)²
d(t) = 400
Average speed/rate of change = distance/time = 64/2 = 32 feet
Average speed/rate of change = distance/time = 400/5 = 80 feet
The rate of change determines the average speed of the ball when it is dropped from the building.
!!
The average rate of change of d(t) from [tex]\( t = 2 \) to \( t = 5 \)[/tex] represents that the ball traveled at an average rate of 112 feet per second during this time interval.
The average rate of change of a function over a given interval is calculated by finding the difference in the function values at the endpoints of the interval and dividing by the difference in the independent variable values. In this case, we want to find the average rate of change of the function [tex]\( d(t) = 16t^2 \) from \( t = 2 \) to \( t = 5 \).[/tex]
First, we find the values of the function at the endpoints of the interval:
- At [tex]\( t = 2 \), \( d(2) = 16(2)^2 = 64 \)[/tex] feet.
- At [tex]\( t = 5 \), \( d(5) = 16(5)^2 = 400 \)[/tex] feet.
Then, we calculate the difference in the function values:
[tex]\[ \text{Difference in } d(t) = d(5) - d(2) = 400 - 64 = 336 \text{ feet} \][/tex]
Next, we calculate the difference in the independent variable values:
[tex]\[ \text{Difference in } t = 5 - 2 = 3 \text{ seconds} \][/tex]
Finally, we find the average rate of change by dividing the difference in d(t) by the difference in t:
[tex]\[ \text{Average rate of change} = \frac{\text{Difference in } d(t)}{\text{Difference in } t} = \frac{336}{3} = 112 \text{ feet/second} \][/tex]