Exercise 1: ABC is a triangle such that AB=5cm. A=50°and AC=5cm 1) Construct the triangle ABC. 2) What is the nature of the triangle ABC? Justity. 3) Calculate ABC and ACB. 4) Draw the height [AH) issued from A to [BC]. 5) a. Calculate BAH and CAH. b. What does [AH] represents for BAC? Justify.​

(a) The maximum value of g(x) = -8[tex]x^2[/tex] + 5x - 2 is 5/16.

(b) The maximum value of h(x) = -(1/3)[tex]x^2[/tex] + 3x is 9/2.

(c) The maximum value of k(x) = 8[tex]x^2[/tex] - 5x - 7 is 5/16.

(d) The maximum value of j(x) = -(71/99)[tex]x^2[/tex] + 210 is 0.

(e) The maximum value of f(x) = 4[tex]x^2[/tex] - 3x + 7 is 3/8.

(f) The maximum value of F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9 is 0/0. We cannot say anything about that function.

The general form of a quadratic function;

f(x) = a[tex]x^2[/tex] + bx + c and a < 0,

Then the function f has a maximum value when x = -b/2a

(a) The function is g(x) = -8[tex]x^2[/tex] + 5x - 2

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -8 and b = 5

Now putting the value at x = -b/2a

x = -5/2(-8)

x = (-5)/(-16)

x = 5/16

(b) The function is h (x) = -(1/3)[tex]x^2[/tex] + 3x

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(1/3) and b = 3

Now putting the value at x = -b/2a

x = (-3)/2(-1/3)

x = (3×3)/2

x = 9/2

(c) The function is k(x) = 8[tex]x^2[/tex] - 5x - 7

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 8 and b = -5

Now putting the value at x = -b/2a

x = -(-5)/(2×8)

x = 5/16

(d) The function is j(x) = -(71/99)[tex]x^2[/tex] + 210

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(71/99) and b = 0

Now putting the value at x = -b/2a

x = 0/2(-(71/99))

x = 0

(e) The function is f(x) = 4[tex]x^2[/tex] - 3x + 7

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 4 and b = -3

Now putting the value at x = -b/2a

x = -(-3)/(2×4)

x = 3/8

(f) The function is F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 0 and b = 0

Now putting the value at x = -b/2a

x = 0/0

We cannot say anything about that function.

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The complete question is given below:

The equation in slope-intercept form that models the total cost of attendance is y = 4487x + 750.

We know lines have various types of equations, the general type is

Ax + By + c = 0, and the equation of a line in slope-intercept form is

y = mx + b.

Where slope = m and b = y-intercept.

the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects

the y-axis at x = 0.

To find the equation in the slope-intercept form,

We'll sum all the annual fees and subtract the scholarship amount and 'x' will represent the number of years while the one-time fee will be our constant term.

So, (6125 + 2112 + 1250 - 5000).

= 4487.

Therefore, The equation in slope-intercept form is,

y = 4487x + 750.

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Answer:

X = 11

Step-by-step explanation:

The total degrees of a line is 180

They gave us 39 and 13x-2

We can get rid of the 39 by subtracting it from 180

180-39=141

That means 13x-2 = 141

Now to find X. Adding 2 on both sides, gets rid of the 2 on the left side

13x = 143

Dividing 13 on both sides, we can get X alone

X = 11

Herb spent 55 hours at the library.

Fractions defines a part of a whole and are represented as a numerical value. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.

Total time spent in the Library = 1/6 + 1/4 + 1/2

Total time spent = 11/12 x 60 mins

Total time spent = 55 hours

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The lines L₁ and L₂ given by the vector equations is perpendicular.

In order to prove that the lines are perpendicular,

L₁: r(t) = (-5+ 3t)i + (3 + t)j

L₂: r(s) = (2+2s)i + (4 - 6s)j

To determine whether the lines L₁ and L2 are parallel, perpendicular, or neither, we need to compare their direction vectors.

The direction vector for L₁ is given by <3,1> and the direction vector for L₂ is given by <2,-6>.

The dot product of the direction vectors is (32) + (1-6) = -8. Since the dot product is zero, the vectors are perpendicular, and therefore the lines are perpendicular.

Therefore, the lines are PERPENDICULAR.

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After the raise, Lisa much will pay $9 in federal income taxes Annually.

For the stated question:

Lisa's salary =  $215.20 a week.

After increment of $40 per week.

$215.20  + $40 = $255.2

She gets 3 allowances and is married.

After raise,

Federal income taxes = $9. (taken from the table)

Thus, After the raise, Lisa much will pay $9 in federal income taxes Annually.

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Answer: 3

Step-by-step explanation: 60 x 5/100 = 3

There were three first-year teachers in the meeting.

In the problem, it is solved that the number of dogs John has is 14

The problem is solved by expression the word problem into a simultaneous equation as follows

let d be the number of dogs and c d number of chickens

d + c = 25 (since all must have a head)

4d + 2c = 78 (2 legs for chicken and 4 legs for dog)

using calculator to solve gives

d = 14 and c = 11

hence the number of dogs is 14

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The piggy bank has 200 dimes and 50 quarters.

To find this, we can set up the following equation:

(number of dimes x $0.10) + (number of quarters x $0.25) = $39.25

We know that the piggy bank contains 250 coins, so we can substitute that in:

(x x $0.10) + (250-x x $0.25) = $39.25

Now we can solve for x, the number of dimes:

x0.10 + (250-x)0.25 = 39.25

x0.10 + 62.5 - x0.25 = 39.25

0.1x + 62.5 - 0.25x = 39.25

-0.15x = -23

x= 153

So there are 153 dimes in the piggy bank, and 250-153 = 97 quarters.

But we know that the piggy bank contains 250 coins, so we can check that 153 dimes + 97 quarters = 250 coins.

The equivalent expression of i⁴¹ is i.

Expressions that are equivalent to the same thing, even when they have distinct appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).

Given:

An expression i⁴¹.

Simplifying,

i⁴¹ = {(i⁴)¹⁰} i

i⁴¹ = {(1)¹⁰} i   {i⁴ = 1}

i⁴¹ = i

Therefore, the equivalent expression is i.

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1) the rate change of function f(x) in given interval =4(b+1)

2)  the rate change of function g(x) in given interval =1

3) the rate change of function j(x) in given interval =[tex]3h^2+9h+9[/tex]

To find out the average rate change of function on given interval is -

Use the graph of f to determine f (a) and f (b) for the provided range [a,b]. Remember that the graph of f's point with the y-coordinate of f (a) and the x-coordinate of an is called f. Similar to that, f ( b ) is the y-coordinate of the point on the f graph whose x-coordinate is b.

Step 2: Calculate the average rate of change of  f over the range [a, b] using the average rate of change formula.

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

1)

[tex]f(x)=4x^2-7 \ on\ [1,b]\\\\f(1)=4-7=-3\\\\f(b)=4b^2-7\\\\the \ rate \ change =\frac{4b^2-7+3}{b-1}\\\\=4(b+1)[/tex]

2)

[tex]p(x)=3x+4 \ on\ [2,2+h]\\\\p(2)=10\\\\p(2+h)=10+h\\\\The\ average\ rate \ change= \frac{10+h-10}{2+h-2}\\\\=1[/tex]

3)

[tex]j(x)=3x^3 \ on \ [1,1+h]\\\\j(1)=3\\\\j(1+h)= 3(1+h)^3\\\\\\so, the\ rate\ change = 3h^2+9h+9[/tex]

The complete Question is: -

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

1. f(x)= 4x^2 - 7 on [1, b]

2. p(x)= 3x+4 on [2,2+h]

3. j(x)= 3x3 on [1, 1+h]

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Answer:

$ 148.75

Step-by-step explanation:

       Principle = $ 425

         Rate = 7%

       Time = 5 yeats

[tex]\sf \boxed{Simple \ Interest = \dfrac{Principle*rate*time}{100}}[/tex]

                           [tex]= \dfrac{425*7*5}{100}\\\\=\dfrac{17*7*5}{4}\\\\\\=\dfrac{595}{4}\\\\=148.75[/tex]

So. the left Riemann sum is 23.74 ft^2.

Given the information in the table:

(a) To use a left Riemann sum with three subintervals to approximate the integral of A(h) with respect to h, we need to divide the interval of integration [0,10] into three equal subintervals. The width of each subinterval is (10-0)/3 = 3.33 ft. The left Riemann sum can be calculated as:

(A(0) * 3.33) + (A(2) * 3.33) + (A(5) * 3.33) = (50.3 * 3.33) + (14.4 * 3.33) + (6.5 * 3.33) = 16.80 + 4.78 + 2.16 = 23.74 ft^2

(b) This approximation represents the total area of the horizontal cross section of the tank up to a specific height h.

(c) The Riemann sum is an underestimate of the actual integral. This is because the Riemann sum uses the left endpoint of each subinterval to approximate the area under the curve, but the actual area under the curve is always greater than or equal to the area of the rectangles.

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Therefore, the 3 patterns are
3,6,9,3,6,9            

Explanation:                      

Given that,

we will carry out the multiplications of succession  counting number by 3 and if there is more than a single digit answer .we add the digits.
we are looking for is a pattern for there single digit numerals.
carryout the plan                  

1x3=3→3
2x3=6→6
3x3=9→9
4x3=12→1+2=3
5x3=15→1+5=6
and so on
continue with some additional at terms.
6x3=18→1+8=9
7x3=21→2+1=3

∵ Therefore the 3 patterns are
3,6,9,3,6,9


A pattern is defined in mathematics as a sequence of repeating objects, shapes, or numbers. A pattern can be associated with any type of event or object. A pattern has a rule that tells us which objects belong to the pattern and which objects do not belong to the pattern.


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When you divide two numbers the answer is called the quotient.

Divide the dividend by the divisor to find the quotient of a division problem.

This may be accomplished by using repeated subtraction or lengthy division.

The quotient of two numbers, two fractions, or two algebraic expressions can be found.

According to the remainder theorem, when a polynomial p(x) is divided by (x – a), the remainder Equals f. (a). This is demonstrated by Euclid’s Division Lemma. Using this, p(x) = q(x) (x – a) + r if q(x) is the quotient and r is the remainder.

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Answer: (-4t - 7) + (3t + 8) cannot be simplified further as the terms have different variables.

(-4r and -7) and (3r and 8) cannot be combined as these terms have different variables.

(-41 and 3r) cannot be combined as these terms have different variables.

Step-by-step explanation:

The predicted value of y when I = 5 is equal to D) 30.

To determine the predicted value of y when I = 5, we can use the equation of the simple linear regression model:

y = b0 + b1x

where b0 and b1 are the regression coefficients and x is the predictor variable.

We can find b1 using the formula:

b1 = r(Sy/Sx)

where r is the sample correlation coefficient, Sy is the standard deviation of the y variable, and Sx is the standard deviation of the x variable.

Plugging in the given values:

b1 = -0.98(8/2) = -4

We can find b0 using the formula:

b0 = ybar - b1*xbar

where ybar and xbar are the mean of the y variable and the x variable respectively.

Plugging in the given values:

b0 = 10 - (-4)*10 = 50

Therefore, the equation of the simple linear regression model is:

y = 50 - 4x

So when I = 5, the predicted value of y is:

y = 50 - 4(5) = 50 - 20 = 30

The closest answer choice is D. 30.

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Answer: Let x be the original cost of the jersey.

We know that the jersey was on sale at a 30% discount, so the sale price is x - (30/100)*x = x(1- 0.3) = 0.7x

We also know that Tyrone spent a total of $83.39 and the hat cost $37.89, so the jersey cost is 83.39 - 37.89 = 45.5

So we can write the equation:

0.7x = 45.5

To find the original cost of the jersey, we can divide both sides by 0.7:

x = 45.5 / 0.7

x = 65

So the original cost of the jersey is $65.

Step-by-step explanation:

The sum of four consecutive integers is,

⇒ 4n + 6

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

We have to given that;

⇒ n = first integer of the four.

Now, Let four consecutive integers with n is first integer of the four are,

⇒ n, n + 1, n + 2, n + 3

Hence, The sum of four consecutive integers is,

⇒ S = n + (n + 1) + (n + 2) + (n + 3)

      = 4n + 6

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The value of dividing 316 by 4 will be 79.

An expression is simply used to show the relationship between the variables that are provided or the data given regarding an information.

In this case, it is vital to note that they have at least two terms which have to be related by through an operator. Some of the mathematical operations that are illustrated in this case include addition, subtraction, etc.

It should be noted that dividing 316 by 4 will be:

= 316 / 4

= 79

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Complete question

Divide 316 by 4

One can infer some information about a line's slope just by looking at its graph, especially in comparison to other lines plotted on the same coordinate plane.

Rise / run = a slope.

The Complete Question is line's slope.

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The required set of positive two-digit even numbers less than 36 that do not contain the digit 2 is 11,13,14,15,16,17,18,19,30,31,33,34,35.

An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Zero is not a fraction or decimal of any number. It is neither positive nor negative.

Given:

Set D is the set of positive two-digit even numbers less than 36 that do not contain the digit 2.

According to the given question, we have

2-digit numbers are numbers that have two digits and they start with the number 10 and end with the number 99.

Starting at 11 and going up the two-digit numbers even numbers less than 36 that do not contain the digit 2.

11,13,14,15,16,17,18,19,30,31,33,34,35.

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Answer:

One Half

Step-by-step explanation:

1/3 = 2/6

2/6 + 1/6 = 3/6 = 1/2

The area of the rectangle is and using the length and breadth

area of rectangle = length × breadth

area of rectangle = 2.5m × (4n +5)

area of rectangle = 10mn + 12.5m

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Question:

The dimensions, in feet, of a rectangle, are represented by expressions, as shown in the diagram below. The area of the rectangle is 10mn + 12.5m. Explain your work.

Step-by-step explanation:

it is a parallelogram.

in comparison to a rectangle one side is shifted a bit, but the endpoints of the shifted side are still positioned the same way relative to each other.

and particularly, their relative offset from each other must be the same as for the endpoints of the other, parallel side.

let's look at

(0, -1) and (2, 7).

their relative offset is given by

x + 2 (from 0 to 2)

y + 8 (from -1 to 7)

the missing point must have the same offset from (-5, -3), as this line is parallel to the one above :

-5 + 2 = -3

-3 + 8 = 5

so, the missing point is (-3, 5).

Another point on the line is  (2, 3)

To find another point on the line the equation of the line will be solved as used to get other points

The slope, m of the lines is calculated the formula

m = (y₀ - y₁) / (x₀ - x₁)

where

m = slope

x₂ and x₁ = points in x coordinates

y₂ and y₁ = points in y coordinates

The slope, m of the linear function is calculated using the points (0, 5), (4, 1)

m = (y₀ - y₁) / (x₀ - x₁)

m = (5 - 1) / (0 - 4)

m = (4) / (-4)

m = -1

equation passing through point (0, 5)

(y - y₁) = m  (x - x₁)

y - 5 = -1 (x - 0)

y - 5 = -x

y = -x + 5 equation of the line

for x = 2

y = -x + 5 = -2 + 5 = 3

the point is (2, 3)

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Answer:

First, find out how many guests have ordered chicken: 240 guests * 3535/10000 = 84 guests

Then subtract that number from the total number of guests to find the number of guests who haven't ordered chicken: 240 guests - 84 guests = 156 guests

Then find how many of those guests have ordered gluten-free meals: 156 guests * 12.5% = 19.5 guests

Then round down the number of guests who have ordered gluten-free meals to the nearest whole number to find the exact number of guests who ordered gluten-free meals: 19 guests

Then subtract that number from the number of guests who haven't ordered chicken to find the number of guests who haven't ordered their meals yet: 156 guests - 19 guests = 137 guests

So 137 people haven't ordered their meals yet.

a) The block-spring system has an elastic potential energy equal to 2.517 joule.

b) The block has speed of approximately 1.513 meters per second.

In this problem we must analyze the energy and kinematics of spring-block system with no irreversibilites (no friction). Part A - Elastic potential energy is determined when spring is entirely deformed:

U = (1 / 2) · k · x²

Where:

If we know that k = 895 N / m and x = 0.0750 m, then the elastic potential energy of the block-spring system is;

U = (1 / 2) · (895 N / m) · (0.0750 m)²

U = 2.517 J

The elastic potential energy of block-spring system is 2.517 joule.

Part B - And the speed can be determined by principle of energy conservation and work-energy theorem:

K = U

Where K is translational kinetic energy, in joules.

(1 / 2) · m · v² = U

v² = 2 · U / m

v = √(2 · U / m)

If we know that U = 2.517 J and m = 2.20 kg, then the speed of the block is:

v = √[2 · (2.517 J) / (2.20 kg)]

v ≈ 1.513 m / s

The speed of the block after leaving the spring is approximately 1.513 meters per second.

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It is proven that the other nth roots of z can be found by multiplying the initial one by the nth roots of unity.

A complex number satisfying the equation [tex]z^{n-1} = 0[/tex] is known as the nth root of unity. There are about n nth roots of a unit, according to the fundamental theorem of algebra. We know that,[tex]1=e^{2\pi i}[/tex]. Then, the nth root is written as [tex]\omega=e^{2\pi i/n}[/tex].

The integer of powers of this root between 0 and n-1 provides the other nth roots. Then, the roots are [tex]1, \omega, \omega^2 ,\cdots \cdots, \omega^{(n-1)}[/tex]. Now, factorize the equation [tex]z^{(n-1)} = 0[/tex] we get the required identity as, [tex]z^{(n-1)} = (z- 1)(z^{(n-1)}+ z^{(n-2)} + \cdots \cdots+ z + 1) = 0[/tex].  

This indicates that the equation [tex]1+\omega+ \omega^2+\cdots \cdots+ \omega^{(n-1)}[/tex] is true for all nth roots of unity other than 1.

Roots of unity also have the great quality of being periodic, which results from the fact that n = 1. It is equal to a power of that power mod n if the power is bigger than n.

When ω is a third root of unity, [tex]\omega^5 + \omega^4 + 1 = \omega^2 + \omega + 1 = 0[/tex].

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The population of Calculus City can be modeled by an exponential function, g(t) = 10(1.3) ^t, where t is the number of years since 2019. Evaluating g (6) means calculating the population of Calculus City after 6 years, which is 10(1.3) = 39.5 thousand people.

(a) The population of Calculus City can be modeled by an exponential function, g(t) = 10(1.3) ^t, where t is the number of years since 2019.

(b) g (6) = 10(1.3) = 46.6 thousand. This means that the population of Calculus City will be 46.6 thousand in the year 2025.

(c) Solving the equation g(t) = 16.9 yields t = log1.3(16.9/10) = 4.7 years. This means that the population of Calculus City will be 16.9 thousand in the year 2023.7.

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