Answer:
Amal's mother was 11.4 years old when Amal was born.
Step-by-step explanation:
Let's start by using variables to represent the ages of each person:
Let A be Amal's ageLet S be Amal's sister's ageLet M be Amal's mother's ageLet F be Amal's father's ageFrom the problem, we know:
S = 0.5AM = 3AF = 4MA + S + M + F = 94Substituting the first three equations into the fourth, we get:
[tex]\sf:\implies A + 0.5A + 3A + 4(3A) = 94[/tex]
Simplifying:
[tex]\sf:\implies A + 0.5A + 3A + 12A = 94[/tex]
[tex]\sf:\implies 16.5A = 94[/tex]
[tex]\sf:\implies A = 5.7[/tex]
So Amal is 5.7 years old. To find the age of Amal's mother when Amal was born, we need to subtract Amal's age from his mother's age:
[tex]\sf:\implies M - A = 3A - A = 2A[/tex]
So Amal's mother was 2A = 2(5.7) = 11.4 years old when Amal was born.
which fraction is equivalent to 0.48 in simplest form?
[A] 12/25
[B] 12/50
[c] 24/50
[D] 48/100
Answer:
0.48 = 48/100
48/100 ÷ 4/4 = 12/25
0.48 = 12/25 =A
Kera took out a 24-month bank loan of $13,000 at an interest rate of 5. 95%. She budgets to pay
$450 per month towards the loan. Write an equation that represents how much total interest
Kera will pay towards the remaining balance of the loan at the end of each year. Let m equal the
number of months paid and r equal the interest charged on the remaining balance
The equation that represents how much total interest Kera will pay towards the remaining balance of the loan at the end of each year is Total Interest Paid = (Remaining Balance) x (Annual Interest Rate) = $422.03.
The equation that represents how much total interest Kera will pay towards the remaining balance of the loan at the end of each year is:
Total Interest Paid = (Remaining Balance) x (Annual Interest Rate)
To calculate the remaining balance after m months, we can use the formula for the present value of an annuity:
Remaining Balance = (Payment per Month) x ((1 - (1 + r)^(-n)) / r)
where r is the monthly interest rate (0.0595 / 12 = 0.004958), n is the total number of months (24), and m is the number of months paid (12, 24, etc.).
Plugging in the given values, we get:
Remaining Balance = 450 x ((1 - (1 + 0.004958)^(-12)) / 0.004958) = $6,752.45
To calculate the annual interest rate, we can use the formula:
Annual Interest Rate = (1 + r)^12 - 1
Plugging in the monthly interest rate, we get:
Annual Interest Rate = (1 + 0.004958)^12 - 1 = 0.0625
Therefore, the equation that represents how much total interest Kera will pay towards the remaining balance of the loan at the end of each year is:
Total Interest Paid = $6,752.45 x 0.0625 = $422.03 (rounded to the nearest cent)
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Would anyone be willing to help me out with a few math questions? I'm up late and could really use the help!
The area of a rooftop can be
expressed as (x + 9)2. The rooftop
is a rectangle with side lengths
that are factors of the expression
describing its area. Which expression
describes the length of one side of
the rooftop?
The expression that describes the length of one side of the rooftop is therefore: x - 9.
What is expression?In mathematics, an expression is a combination of one or more variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. An expression can be as simple as a single variable or constant, or it can be a more complex combination of variables and operations.
Here,
The expression for the area of the rooftop is (x + 9)², where x is a variable representing the length of one side of the rectangle. To find the factors of this expression, we can expand it using the identity (a+b)² = a² + 2ab + b².
Expanding (x + 9)², we get:
(x + 9)² = x² + 18x + 81
Now, we need to find the factors of this expression that are also factors of the length of the sides of the rectangle. Since the sides of the rectangle must have a common factor of x, we can factor out x from the expression:
x² + 18x + 81 = x(x + 18) + 81
The factors of (x + 9)² are x(x + 18) + 81, (x + 9)(x + 9), (x - 9)(x - 9), and -(x + 9)(x + 9).
Since we are looking for factors that represent the length of one side of the rooftop, we can eliminate the negative factor and the factor (x + 9)(x + 9), since the sides of a rectangle must be positive.
That leaves us with x(x + 18) + 81 and (x - 9)(x - 9).
The expression describes the length of one side of the rooftop: x - 9
This is because the sides of a rectangle must be positive, and (x - 9) is a factor of (x + 9)² that represents a positive length.
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Find the smallest number of terms of the series ∑ n = 1 (-1)^n+1/2^n you need to be certain that the partial sum Sn is within 1/100 of the sum.n=2 n=4 n=6 n=8 n=7
We need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.
We want to find the smallest value of n such that the absolute value of the difference between the sum of the first n terms and the sum of the entire series is less than 1/100.
The sum of the first n terms of the series is given by:
Sn = ∑_(k=1[tex])^n[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]
We can write the sum of the entire series as:
S = ∑_(k=[tex]1)^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex]
The absolute value of the difference between the sum of the first n terms and the sum of the entire series is:
|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] [tex](-1)^(k+1)/2^k|[/tex]
We want to find the smallest value of n such that |S - Sn| < 1/100.
Let's start by evaluating the sum of the series:
S = ∑_(k=1) (-1[tex])^(k+1)[/tex]/[tex]2^k[/tex] = 1/2 - 1/4 + 1/8 - 1/16 + ...
This is a geometric series with first term a = 1/2 and common ratio r = -1/2. The sum of the series is:
S = a/(1-r) = (1/2)/(1+1/2) = 1/3
Now we can write:
|S - Sn| = |∑_(k=n+1[tex])^∞[/tex] (-1[tex])^(k+1)[/tex]/[tex]2^k|[/tex] <= 1/[tex]2^(n+1)[/tex]
The last inequality is true because the terms of the series are decreasing in absolute value, and we are summing an infinite number of terms.
Therefore, we need to find the smallest value of n such that 1/2^(n+1) < 1/100. This gives:
n+1 > log2(100)
n > log2(100) - 1
n > 6.64
The smallest integer value of n that satisfies this inequality is n = 7.
Therefore, we need at least 7 terms of the series to be certain that the partial sum Sn is within 1/100 of the sum.
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Find f'(-2) for f(x) = ln((x^4 + 5)^2). Answer as an exact fraction or round to at least 2 decimal places.
Using the chain rule, we have: f'(x) = 2ln(x^4 + 5) * 2(x^4 + 5)^1 * 4x^3
f'(x) = 16x^3 * ln(x^4 + 5) * (x^4 + 5)
To find f'(-2), we plug in -2 for x:
f'(-2) = 16(-2)^3 * ln((-2)^4 + 5) * ((-2)^4 + 5)
f'(-2) = -128 * ln(21) * 21
f'(-2) ≈ -599.92 (rounded to 2 decimal places)
Therefore, f'(-2) is approximately -599.92.
To find f'(-2) for the function f(x) = ln((x^4 + 5)^2), we will first find the derivative of the function, and then evaluate it at x = -2.
1. Differentiate the function using the chain rule:
f'(x) = (d/dx) ln((x^4 + 5)^2) = (1/((x^4 + 5)^2)) * (d/dx) ((x^4 + 5)^2)
2. Differentiate the inner function:
(d/dx) ((x^4 + 5)^2) = 2(x^4 + 5) * (d/dx) (x^4 + 5) = 2(x^4 + 5) * (4x^3)
3. Combine the derivatives:
f'(x) = (1/((x^4 + 5)^2)) * (2(x^4 + 5) * (4x^3)) = (8x^3(x^4 + 5))/((x^4 + 5)^2)
4. Evaluate the derivative at x = -2:
f'(-2) = (8(-2)^3((-2)^4 + 5))/((-2)^4 + 5)^2 = (-128(21))/(21^2) = -128/21
So, f'(-2) is -128/21 as an exact fraction.
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Can someone please answer the question below (Level: Year 8 (7th Grade) ) about algebra equations?
Thanks ^^
help please
Use substitution to find the indefinite integral. x² - 2x 3 X - S dx x4 - 4x + 4 - X x - 2x 4 - 4x + 4 dx=
To solve this problem, we can use substitution. Let's set u = x-2. Then, du/dx = 1 and dx = du. Using this substitution, we can rewrite the integral as:
∫ (u+2)² - 2(u+2) 3 (u+2) - Su du
Expanding the terms inside the integral, we get:
∫ (u² + 4u + 4) - 2(u+2)³ (u+2) - Su du
Simplifying, we get:
∫ u⁴ - 4u³ + 4u² - u³ + 6u² - 12u - u² + 6u - 9 du
Combining like terms, we get:
∫ u⁴ - 5u³ + 9u² - 6u - 9 du
Now, we can integrate each term separately using the power rule of integration:
∫ u⁴ - 5u³ + 9u² - 6u - 9 du = (1/5)u⁵ - (5/4)u⁴ + (9/3)u³ - 3u² - 9u + C
Substituting back u = x-2, we get:
(1/5)(x-2)⁵ - (5/4)(x-2)⁴ + (3)x³ - 3(x-2)² - 9(x-2) + C
Therefore, the indefinite integral of x² - 2x 3 X - S dx x⁴ - 4x + 4 - X x - 2x⁴ - 4x + 4 dx is (1/5)(x-2)⁵ - (5/4)(x-2)⁴ + (3)x³ - 3(x-2)² - 9(x-2) + C.
Hi! I'd be happy to help you with your integration problem. To find the indefinite integral using substitution, let's first rewrite the given integral:
∫(x² - 2x) / (x⁴ - 4x² + 4) dx
Now, let's perform substitution:
Let u = x² - 2x
Then, du/dx = 2x - 2
And also let v = x⁴ - 4x² + 4
Then, dv/dx = 4x³ - 8x
We need to find du in terms of dx, so:
du = (2x - 2) dx
Now, we can rewrite the integral in terms of u and v:
∫(u) / (v) (du / (2x - 2))
Now we can integrate:
(1/2) ∫(u) / (v) du
Unfortunately, this integral does not have a straightforward elementary antiderivative. However, you can use numerical integration methods or special functions to approximate the indefinite integral if necessary.
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At Olivia's Hats, 90% of the 80 hats are baseball caps. How many baseball caps are there?
Let g(x) be continuous with g(0) = 3. g(1)
8, g(2) = 4. Use the Intermediate Value Theorem to ex-
plain why s(x) is not invertible.
The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a,b], and if M is any number between f(a) and f(b), then there exists at least one number c in the interval [a,b] such that f(c) = M.
In this case, we are given a continuous function g(x) with g(0) = 3, g(1) = 8, and g(2) = 4. Let s(x) be the inverse of g(x), which means that s(g(x)) = x for all x in the domain of g(x).
Suppose s(x) is invertible. Then for any y in the range of g(x), there exists a unique x such that g(x) = y, and therefore s(y) = x. In particular, let y = 5, which is between g(1) = 8 and g(2) = 4. By the Intermediate Value Theorem, there exists a number c in the interval [1,2] such that g(c) = 5.
However, this means that s(5) is not well-defined, since there are two values of x (namely c and s(5)) that satisfy g(x) = 5. Therefore, s(x) is not invertible.
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists a number c in the interval [a, b] such that f(c) = k.
Let g(x) be continuous with g(0) = 3, g(1) = 8, and g(2) = 4. Since g(x) is continuous, the Intermediate Value Theorem applies. However, to show that s(x) is not invertible, we need to show that g(x) is not one-to-one.
Notice that g(0) = 3 and g(2) = 4, with g(1) = 8 in between. This means that there must exist a point c1 in the interval (0, 1) such that g(c1) = 4, and another point c2 in the interval (1, 2) such that g(c2) = 3, due to the Intermediate Value Theorem.
Since g(c1) = g(c2) = 4 and c1 ≠ c2, g(x) is not one-to-one. Therefore, its inverse function s(x) does not exist, and s(x) is not invertible.
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How many zeros are in the product 50 x 6,000
The number of zeros are in the product of the number 50 and 6000 is 50 x 6000 = 300,000 are five.
Integers, natural numbers, fractions, real numbers, complex numbers, and quaternions are examples of typical special instances where it is possible to define the product of two numbers or the multiplication of two numbers.
A product is the outcome of multiplication in mathematics, or an expression that specifies the elements (numbers or variables) to be multiplied.
The commutative law of multiplication states that the result is independent of the order in which real or complex numbers are multiplied. The result of a multiplication of matrices or the elements of other associative algebras typically depends on the order of the components. For instance, matrix multiplication and multiplication in general in other algebras are non-commutative operations.
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equation of a line with slope m=−2/5 that contains the point (10,−5).
Answer:
y = (-2/5)x+b
Step-by-step explanation:
First plug these into the y=mx+b equation:
-5 = (-2/5)(10)+b.
Then solve for b:
-5 = -4+b
Add 4 to both sides:
-1 =b.
Therefore, the equation of the line is y = (-2/5)x+b. You can also double check this by plugging 10 into the equation we just obtained.
Clarissa is cutting construction paper into rectangles for a project. She needs to cut one rectangle that is 10 inches × 14 1/2
inches. She needs to cut another rectangle that is 10 1/4
inches by 10 1/2 inches. How many total square inches of construction paper does Clarissa need for her project?
Clarissa needs a total of 251.25 square inches of construction paper for her project.
To find the total area of construction paper needed, we need to find the area of each rectangle and add them together.
The first rectangle has an area of 10 inches × 14.5 inches = 145 square inches.
The second rectangle has an area of 10.25 inches × 10.5 inches = 107.625 square inches.
Adding these two areas together, we get a total of 145 + 107.625 = 252.625 square inches.
Therefore, Clarissa needs a total of 251.25 square inches (rounded to two decimal places) of construction paper for her project.
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Help with the problem in photo
The measure of the unknown arc is 76°
What is arc angle relationship?An arc is a smooth curve joining two endpoints. It can also be defined as a portion of a circle.
Arc angle relationship states that If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Therefore angle 30° = 1/2 ( x-61)
30 = 1/2( x -61)
15 = x -61
x = 15 + 61
x = 76°
therefore the measure of the unknown arc is 76°
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Please help me with this math problem!!! Will give brainliest!!
The average price of milk in 2018 was 6.45 dollars per gallon.
The average price of milk in 2021 was 189.95 dollars per gallon.
How to calculate the priceThe given function is: Price = 3.55 + 2.90(1 + x)³
In order to find the average price of milk in 2018, we need to set x = 0:
Price in 2018 = 3.55 + 2.90(1 + 0)³ = 3.55 + 2.90(1) = 6.45 dollars per gallon
Price in 2021 = 3.55 + 2.90(1 + 3)³ = 3.55 + 2.90(64) = 189.95 dollars per gallon
Hence, the average price of milk in 2021 was 189.95 dollars per gallon.
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What adds to +10 and multiplys to 290
A museum groundskeeper is creating a simicircular stauary garden with a diameter of 38 feet there will be a fence around the garden the fencing cost $9.25 per linear foot . About how much will the fencing cost although? Round to the nearest hundredth use 3.14 for n the fencing will cost about $
The amount for the fencing cost is $903. 36
How to determine the valueFrom the information given, we have that the shape of the garden is semi -circle.
Now, the formula that is used for calculating the circumference of a semicircle is expressed as;
C = πr + 2r
Given that the parameters of the equation are;
C is the circumference of the semicircler is the radius of the semicircleFrom the information given,
Substitute the values, we have;
Circumference = 3.14(19) + 2(19)
expand the bracket
Circumference = 59. 66 + 38
Add the values
Circumference = 97. 66 feet
Then,
if 1 feet = $9.25
Then, 97. 66 feet = x
x = $903. 36
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Calculate the discount period for the bank to wait to receive its money. (Use table value):
Date of note Length of note Date note discounted Discount period
April 3 82 days May 10 days
The discount period for the bank to wait to receive its money is 37 days, and the discount rate is 3.5%.
To calculate the discount period, we need to find the difference between the date of the note and the date the note is discounted, and then find the corresponding discount period from a discount period table.
Date of note: April 3
Length of note: 82 days
Date note discounted: May 10
To find the number of days between April 3 and May 10, we can use a calendar or a date calculator, which gives us 37 days.
Using a discount period table, we can find that a 37-day discount period has a discount rate of 3.5%.
Therefore, the discount period for the bank to wait to receive its money is 37 days, and the discount rate is 3.5%.
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Khloe is a teacher and takes home 90 papers to grade over the weekend. She can
grade at â rate of 10 papers per hour. Write a recursive sequence to represent how
many papers Khloe has remaining to grade after working for n hours.
The recursive sequence representing how many papers Khloe has remaining to grade after working for n hours is given by a_n = a_{n-1} - 10, where a_0 = 90.
Let a_n denote the number of papers Khloe has remaining to grade after n hours of work. After the first hour of work, she will have 90 - 10 = 80 papers remaining. Therefore, we have a_1 = 90 - 10 = 80.
After the second hour of work, she will have a_2 = a_1 - 10 = 80 - 10 = 70 papers remaining. Similarly, after the third hour of work, she will have a_3 = a_2 - 10 = 70 - 10 = 60 papers remaining.
In general, after n hours of work, Khloe will have a_n = a_{n-1} - 10 papers remaining to grade. This is a recursive sequence, where the value of a_n depends on the value of a_{n-1}. The initial value of a_0 is given as 90, since she starts with 90 papers to grade. Therefore, the recursive sequence is given by a_n = a_{n-1} - 10, where a_0 = 90.
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Find a formula for the sum of n terms. Use the formula to find the limit as n = [infinity].
lim ∑ ( 6 + i/n) (2/n)
To find a formula for the sum of n terms, we need to first write out the first few terms of the series and look for a pattern:
n=1: (6+1/1) (2/1) = 14
n=2: (6+1/2) (2/2) + (6+2/2) (2/2) = 16
n=3: (6+1/3) (2/3) + (6+2/3) (2/3) + (6+3/3) (2/3) = 17 1/3
n=4: (6+1/4) (2/4) + (6+2/4) (2/4) + (6+3/4) (2/4) + (6+4/4) (2/4) = 18
From this, we can see that the nth term is given by (6+i/n) (2/n). To find the sum of n terms, we simply add up all of the terms from i=1 to i=n:
∑ (6+i/n) (2/n) = (2/n) ∑ (6+i/n)
Using the formula for the sum of an arithmetic series, we get:
∑ (6+i/n) = n/2 (6 + (6+n)/n)
Substituting this back into our expression for the sum of n terms, we get:
∑ (6+i/n) (2/n) = (2/n) * (n/2) * (6 + (6+n)/n) = 6 + (6+n)/n
Taking the limit as n approaches infinity, we get:
lim (6 + (6+n)/n) = 6 + lim ((6+n)/n) = 6 + 1 = 7
Therefore, the limit of the given series as n approaches infinity is 7.
To find the formula for the sum of n terms, we will use the concept of Riemann sums. Given the expression you provided, it appears that you are trying to compute the limit of the Riemann sum as n approaches infinity, which will give you the integral of the function.
Expression: lim (n→∞) ∑ (6 + i/n) (2/n)
First, let's rewrite the Riemann sum in integral form:
∫(6 + x)dx
Now we need to find the integral of the function and evaluate it over a specific interval. However, you haven't provided the interval, so I'll assume it is [a, b].
∫(6 + x)dx evaluated from a to b will give us the formula for the sum of n terms:
F(x) = 6x + (1/2)x^2
Now, evaluate F(x) over the interval [a, b]:
F(b) - F(a) = [6b + (1/2)b^2] - [6a + (1/2)a^2]
This is the formula for the sum of n terms. To find the limit as n approaches infinity, you will need to provide the specific interval [a, b]. Otherwise, the limit cannot be determined without further information.
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Try Again ml A patient is being treated for a chronic illness. The concentration C(x) (in of a certain medication in her bloodstream x weeks from now is approximated by the following equation 28² 2x+7 CG) - x²–2x+2 Complete the following (a) Use the ALEKS.chine calculator to find the value of x that maximizes the concentration Then give the maximum concentration, Round your answers to the nearest hundredth Value of that maximizes concentration 119 weeks Maximum concentration: 7:19 ml (b) Complete the following sentence For very large, the concentration appears to increase without bound.
The value of x that maximizes the concentration is 119 weeks, and the maximum concentration is approximately 7.19 ml.
How to find maximum concentration?Based on the provided equation, the concentration C(x) is a quadratic function of x with a negative coefficient for the quadratic term, which means that it has a maximum point.
(a) To find the value of x that maximizes the concentration, we can take the derivative of the concentration function with respect to x, set it equal to zero, and solve for x. The derivative of C(x) is:
C'(x) = 56x + 7
Setting C'(x) equal to zero, we get:
56x + 7 = 0
Solving for x, we get:
x = -7/56 = -0.125
However, x represents the number of weeks from now, which cannot be negative. Therefore, the maximum concentration occurs at the endpoint of the interval we are considering, which is x = 119 weeks.
To find the maximum concentration, we can substitute x = 119 into the concentration function:
C(119) = 28²(2119)+7 - 119²-2119+2 ≈ 7.19 ml
So, the value of x that maximizes the concentration is 119 weeks, and the maximum concentration is approximately 7.19 ml.
(b) For very large values of x, the quadratic term (-x²) dominates the concentration function, and the concentration appears to decrease without bound.
This is because the negative quadratic term becomes much larger than the linear term (2x) and the constant term (2), causing the concentration to become more and more negative as x increases.
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A recipe requires only blueberries and strawberries. This list shows the amounts required for 1/4 of the whole recipe:
1/2 cup blueberries
2/5 cup of strawberries
What is the number of cups of blueberries and the number of cups of strawberries required for the whole recipe?
a) 1/8 cup of blueberries and 1/10 cup of strawberries
b) 1/8 cup of blueberries and 1 3/5 cups of strawberries
c) 2 cups of blueberries and 1/10 cup of strawberries
d) 2 cups of blueberries and 1 3/5 cups of strawberries
A
Either divide each by one fourth or multiply each by 0.25. Then turn the answer to a fraction.
Mathematics/g12
nsc
march 2021
question1
a group of workers is erecting a fence around a nature reserve. they store their tools in a shed at
the entrance to the reserve. each day they collect their tools and erect 0,8km of new fence. they
then lock up their tools in the shed and return the next day.
1.1 if the fence takes 40 days to erect, how far would the workers have travelled in total?
(4)
The workers would have traveled a total of 32 km while erecting the fence over 40 days.
To determine the total distance the workers traveled while erecting the fence, we can use the following terms: daily distance, number of days, and total distance.
Step 1: Determine the daily distance traveled.
The workers erect 0.8 km of new fence each day.
Step 2: Determine the number of days it takes to erect the fence.
It takes 40 days to erect the fence.
Step 3: Calculate the total distance traveled.
To find the total distance, multiply the daily distance (0.8 km) by the number of days (40).
Total distance = Daily distance × Number of days
Total distance = 0.8 km × 40
Total distance = 32 km
So, the workers would have traveled a total of 32 km while erecting the fence over 40 days.
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What is the area of EFG
Answer:
A = 28 units²
Step-by-step explanation:
the area (A) of a triangle is calculated as
A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )
here b = FG = 8 and h = FE = 7 , then
A = [tex]\frac{1}{2}[/tex] × 8 × 7 = 4 × 7 = 28 units²
WHY CANT YOU JUST GIVE AN ANSWER WITHOUT MAKING THE PERSON PAY I JUST WANT A EXPLANAITION FOR A QUESTION STILL YOUR MAKING ME PAY ME JUST FOR A ANSWER AND A SIMPLE EXPLANAITION YOUR ADDS ALWAYS FREEZE SO NOW K HAVE TO PAY? OH MY GOD EVERY OTHER WEBSITE DOES THE SAME THING WHY DO YOU DO THAT WITH THE REST IM JUST A GUEST oop sorry caps lock.
steal it
Step-by-step explanation:
Which equation represents a line passing through the points (0, 1) and (2, -3)?
The equation of the line passing through the points (0, 1) and (2, -3) is y = -2x + 1.
What is the equation of the line?The formula for equation of line is expressed as;
y = mx + b
Where m is slope and b is y-intercept.
First, we determine the slope of the line.
Given the two points are (0, 1) and (2, -3).
We can find the slope of the line by using the slope formula:
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values, we get:
m = (-3 - 1) / (2 - 0)
m = -4 / 2
m = -2
Using the point-slope form, plug in one of the given points and slope m = -2 to find the equation of the line.
Let's use the point (0, 1):
y - y₁ = m(x - x₁)
y - 1 = -2(x - 0)
y - 1 = -2x
y = -2x + 1
Therefore, the equation of the line is y = -2x + 1.
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Find dx/dt at x = -5 if y = -5x^2 + 2 and dy/dt = - 4.
dx/dt = ?
x = -5, dx/dt is equal to -2/25.
To find dx/dt, we need to use the chain rule of differentiation.
We know that dy/dt = -4 and we have the equation y = -5x^2 + 2.
Taking the derivative of both sides with respect to t, we get:
dy/dt = d/dt (-5x^2 + 2)
Using the chain rule, we can write this as:
dy/dt = (-10x) (dx/dt)
Now, we can plug in x = -5 and dy/dt = -4:
-4 = (-10(-5)) (dx/dt)
Simplifying, we get:
-4 = 50 (dx/dt)
Dividing both sides by 50, we get:
dx/dt = -4/50
Simplifying further, we get:
dx/dt = -2/25
Therefore, at x = -5, dx/dt is equal to -2/25.
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Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator. In(8x2 – 72x + 112) Enter the
The fully expanded expression using logarithm properties is:
[tex]In(8x^2 - 72x + 112) = 2.079 + In(x - 2) + In(x –-7)[/tex]
How to expand an expression?To expand the given expression[tex]In(8x^2 - 72x + 112)[/tex], we can use the following logarithmic properties:
Product Rule: [tex]logb (xy) = logb x + logb y[/tex]
Quotient Rule: [tex]logb (x/y) = logb x - logb y[/tex]
Power Rule:[tex]logb (x^a) = a logb x[/tex]
We can first factor out a common factor of 8 from the expression inside the logarithm:
[tex]In(8x^2 - 72x + 112) = In[8(x^2 - 9x + 14)][/tex]
Using the distributive property, we can expand the expression inside the logarithm:
[tex]In[8(x^2 - 9x + 14)] = In(8) + In(x^2 - 9x + 14)[/tex]
Now, we need to expand the second logarithm. We notice that the expression inside the logarithm can be factored as follows:
[tex]x^2 - 9x + 14 = (x - 2)(x - 7)[/tex]
Using the product rule, we can write:
[tex]In(x^2 - 9x + 14) = In[(x - 2)(x - 7)][/tex]
[tex]= In(x - 2) + In(x - 7)[/tex]
Putting all the pieces together, we get:
[tex]In(8x^2 - 72x + 112) = In(8) + In(x^2 - 9x + 14)[/tex]
[tex]= In(8) + In(x -2) + In(x - 7)[/tex]
Finally, we can simplify the numerical expression In(8) by using the fact that ln(e) = 1:
[tex]In(8) = ln(8)/ln(e) = 2.079/1 = 2.079[/tex]
So the fully expanded expression using logarithm properties is:
[tex]In(8x^2 - 72x + 112) = 2.079 + In(x - 2) + In(x -7)[/tex]
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2
A rhombus has a perimeter of 136 inches and one diagonal of 60 in.
What is the length of the other diagonal?
Find the area of the rhombus. .
Diagonal =
in
Area =
in2
The area of the rhombus is 1140 square inches.
Let the side length of the rhombus be "a" and let the length of the other diagonal be "d".
Since a rhombus has all sides congruent, the perimeter is given by:
4a = 136
Simplifying, we get:
a = 34
We can use the formula for the area of a rhombus:
Area = (diagonal 1 x diagonal 2)/2
Substituting the given values:
Area = (60 x d)/2
Area = 30d
Now we can substitute the value of "a" in terms of "d" into the formula for the length of the diagonal:
d = √(a² + b²)
d = √(34² + b²)
d = √(1156 + b²)
We also know that the perimeter of the rhombus is given by:
4a = 136
Substituting the value of "a" we found earlier:
4(34) = 136
So the length of the other diagonal can be found by subtracting the length of the given diagonal from the perimeter, and dividing by 2:
d = (136 - 60)/2 = 38
Therefore, the length of the other diagonal is 38 inches.
To find the area, we can substitute the value we found for "d" into the formula we derived earlier:
Area = 30d = 30(38) = 1140
So the area of the rhombus is 1140 square inches.
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The arrow on this spinner is equally likely to land on each section. the arrow is spun 72 times. how many times do you expect the arrow to land on 4?
we know that the spinner has an equal chance of landing on each section. Since there are a total of six sections on the spinner, we can assume that the probability of the arrow landing on any one section is 1/6 or approximately 0.1667.
Now, if the arrow is spun 72 times, we can use this probability to calculate the expected number of times the arrow will land on 4. To do this, we simply multiply the probability by the number of spins, as follows:
Expected number of times arrow lands on 4 = Probability of arrow landing on 4 x Number of spins
Expected number of times arrow lands on 4 = 0.1667 x 72
Expected number of times arrow lands on 4 = 12
So, we can expect the arrow to land on 4 approximately 12 times out of 72 spins. Of course, this is just an expected value, and the actual number of times the arrow lands on 4 may vary from this value due to random chance.
In summary, if we assume that the arrow on the spinner is equally likely to land on each section, and it is spun 72 times, we can expect the arrow to land on 4 approximately 12 times based on probability calculations.
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