The slope-intercept form equation for the word problem is [tex]\( y = \frac{3}{4}x + 1 \).[/tex]
Let's consider a word problem where a local gym charges a $1 sign-up fee and $3 for every hour of use. If we let [tex]\( x \)[/tex] represent the number of hours a member uses the gym and [tex]\( y \)[/tex] represent the total cost, we can write an equation in slope-intercept form.
The slope-intercept form of a linear equation is given by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
In this scenario, the slope [tex]\( m \)[/tex] is the cost per hour, which is $3. The y-intercept [tex]\( b \)[/tex] is the initial sign-up fee, which is $1.
Therefore, the equation representing the total cost [tex]\( y \)[/tex] as a function of the number of hours [tex]\( x \)[/tex] used at the gym is:
[tex]\[ y = 3x + 1 \][/tex]
However, to make the equation more precise and to express the cost per hour as a fraction, we can write $3 as [tex]\( \frac{3}{1} \)[/tex] and then multiply both the numerator and the denominator by 4 to get [tex]\( \frac{3 \times 4}{1 \times 4} = \frac{12}{4} \)[/tex]. Simplifying the fraction [tex]\( \frac{12}{4} \)[/tex] gives us [tex]\( \frac{3}{4} \)[/tex]per hour.
Thus, the final slope-intercept form equation for the word problem is:
[tex]\[ y = \frac{3}{4}x + 1 \][/tex]
An arhitect desings a rectangular flower garden such that the width is exactly two-thirds of the length. If 280 feet of antique picket fencing are to be used to enclose the garden, find the dimensions of the garden.
The derivative of f(x)=(x^4/3)-(x^5/5) attains its maximum at x= ? ...?
The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.
Further explanation:
Given:
The function is [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]
Explanation:
The given function is [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]
Differentiate the above equation with respect to x.
[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right) &= \frac{d}{{dx}}\left( {\frac{{{x^4}}}{3} - \frac{{{x^5}}}{5}} \right)\\&= \frac{{4{x^3}}}{3} - \frac{{5{x^4}}}{5}\\&= \frac{{4{x^3}}}{3} - {x^4}\\\end{aligned}[/tex]
Again differentiate with respect to x.
[tex]\begin{aligned}\frac{{{d^2}}}{{d{x^2}}}f\left( x \right) &= \frac{{{d^2}}}{{d{x^2}}}\left( {\frac{{4{x^3}}}{3} - {x^4}} \right)\\&=\frac{{3 \times 4{x^2}}}{3} - 4{x^3}\\&= 4{x^2} - 4{x^3}\\\end{aligned}[/tex]
Substitute the first derivative equal to zero.
[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right)&= 0\\\frac{{4{x^3}}}{3} - {x^4}&= 0\\\frac{{4{x^3}}}{3} &= {x^4}\\\frac{4}{3}&= \frac{{{x^4}}}{{{x^3}}}\\\frac{4}{3}&= x\\\end{aligned}[/tex]
The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of derivatives
Keywords: Derivative, attains, maximum, value of x, function, differentiate, minimum value.
write 2/5 as a decimal
What is the lateral surface area of a cylindrical-shaped candle with a radius of 4 cm and a height of 14 cm? (Use 3.14 for pi .)
A.
452.16 sq cm
B.
56 sq cm
C.
351.68 sq cm
D.
1,406.72 sq cm
Answer:
it is c
Step-by-step explanation:
A box with a square base of side a is three times higher than its width. Express the volume V of the box as a function of a.
V(a) = ?
The volume V of a box with a square base of side a and a height that is three times its width is expressed as the function V(a) = 3a³.
To express the volume V of a box as a function of its base side length a, with the height being three times its width, we can use the formula for the volume of a rectangular prism, which is the product of its length, width, and height. As the base is a square with side a, both the length and the width of the base are a. Given that the box's height is three times its width (or length, since it is a square), the height will be 3a.
Therefore, the volume V of the box as a function of a is V(a) = a² × (3a) = 3a³.
Which expression is equal to the number of grams (g) in 2.43 kilograms (kg)?
Answer:
the answer would be D
Step-by-step explanation:
Trevor solved the system of equations below. What mistake did he make in his work?
2x + y = 5
x − 2y = 10
y = 5 − 2x
x − 2(5 − 2x) = 10
x − 10 + 4x = 10
5x − 10 = 10
5x = 0
x = 0
2(0) + y = 5
y = 5
He should have substituted 5 + 2x
He combined like terms incorrectly, it should have been 4x instead of 5x
He subtracted 10 from the right side instead of adding 10 to the right side
He made no mistake ...?
Answer:
Give the guy above me Brainlyist
Step-by-step explanation:
500/100 in simplest form
Write the quadratic function in vertex form.
y = x2 - 2x + 5
Answer:
[tex]y=(x-1)^{2}+4[/tex]
Step-by-step explanation:
To write this quadratic function in vertex form, which is the explicit form of the parabola, we have to complete the square in the expression.
First, we have to take the coefficient of the linear term and find the squared power of its half:
[tex](\frac{b}{2} )^{2}=(\frac{2}{2} )^{2}=1[/tex]
Then, we add and subtract this number in the quadratic expression:
[tex]y=x^{2}-2x+5+1-1[/tex]
Now, we use the three terms that can be factorize as the squared power of a binomial expression:
[tex]y=(x^{2}-2x+1)+5-1[/tex]
Then, we find the square root of the first term and third term, and we form the squared power:
[tex]y=(x-1)^{2}+4[/tex]
Now, this vertex form is explicit, because it says from the beginning what's the coordinates of the vertex, which is: [tex](1;4)[/tex], as minimum, because the parabola is concave up.
In vertex form, the quadratic equation is written as;
⇒ y = (x - 1)² + 4
What is Quadratic equation?An algebraic equation with the second degree of the variable is called an Quadratic equation.
We have to given that;
The quadratic equation is,
⇒ y = x² - 2x + 5
Now, We can write in vertex form as;
⇒ y = x² - 2x + 5
⇒ y = x² - 2x + 1 + 4
⇒ y = (x - 1)² + 4
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(AB)^2 + (BC)^2 = (AC)^2
BC = in.
what line is parallel to x-3y=24
whats the difference of 1cm and 1cm2
Final answer:
1 cm is a unit of length equal to [tex]10^{-2}[/tex], meter while [tex]1 cm^2[/tex]is a unit of area equal to 10^{-4} square meters.
Explanation:
The difference between 1 cm and 1 cm2 is that 1 cm is a measure of length, while 1 cm2 is a measure of area. To clarify, 1 cm is equivalent to 10-2 meters (or 0.01 meters), and it represents a one-dimensional measurement. In contrast, 1 cm2 is the area of a square with 1 cm long sides. When converting to square meters (m2), remember that the conversion factor for area is the square of the conversion factor for length. Therefore, 1 cm2 equals (10-2 m)2 or 10-4 m2.
When the new book Units of Fire was released, Jonathan decided to read it in one sitting for his book report. He started at 11:00 a.m. on Saturday morning and read until 8:00 p.m. that night, for a total of 9 hours. For school, however, he needed to record the exact number of minutes he spent reading.
How many minutes did he read?
...?
The total time in minutes that he spent in reading is:
540 minutes.
Step-by-step explanation:We are asked to find the total number of minutes Jonathan spent on reading on Saturday.
It was given that he read for a total of 9 hours.
Hence, we will convert this time to minutes by using the conversion:
As 1 hour= 60 minutes.
Hence, 9 hours= 9×60 minutes= 540 minutes.
Hence, the total time Jonathan spent in reading on Saturday is:
540 minutes.
U have 5 pens. u get 5 more pens. how many pens do u have now
What did the people learn about the banks during this fireside chat?
If the hypotenuse and an acute angle of one right triangle are equal to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent by the HA theorem. If a leg and an acute angle of one right triangle are equal to the corresponding parts of another right triangle, then the triangles are congruent by the _______ theorem.
AL
LA
HL
b?
State the importance of examining a function analytically as well as graphically.?
Answer:
Sometimes it is good to examine function analytically and sometimes it is good to examine function graphically it also depends on the nature of the function.
Moreover, when the function is examined graphically it does not shows the discontinuity of a single point, but it can be examined clearly by analytically.
Also, plotting the graph function is apparently clear by one view but it is not with examining the function analytically.
If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then ...?
Determine if the ordered pair is a solution of the equation.
Is (2,4) a solution of y = 10 -3x?
Question 3 options:
True
False
What is 8 1/4 devided by 1/2?
determine the exact value of the trig ratios:
cos(13pi/4)
cot(11pi/2)
sec(5pi/3)
This isn't the only trick you'll need, but it will help you get these expressions in terms you'll be able to handle more easily. For example, for the cot example above: cot(-pi/2) = cos(-pi/2)/sin(-pi/2) = 0/-1 = 0
The exact values of the trig ratios are sqrt(2)/2, 0, and 2, for cos(13pi/4), cot(11pi/2), and sec(5pi/3) respectively.
To determine the exact value of the trig ratios, we can use the unit circle and trigonometric identities.
cos(13pi/4):
The angle 13pi/4 represents a full circle plus one revolution, so the cosine value is the same as cos(pi/4).
cos(pi/4) = sqrt(2)/2
cot(11pi/2):
The angle 11pi/2 represents 5 full circles plus a half revolution, so the cotangent value is the same as cot(pi/2).
cot(pi/2) = 0
sec(5pi/3):
The angle 5pi/3 represents 2 full circles plus two-thirds of a revolution, so the secant value is the same as sec(pi/3).
sec(pi/3) = 2
A laptop computer is purchased for $1550 . after each year, the resale value decreases by 25% . what will the resale value be after 3 years?
Before beginning voice lessons, Justin already knew how to sing 1 piece, and he expects to learn 2 new pieces during each week of lessons. Write an equation that shows the relationship between the number of weeks x and the number of pieces learned y. Then Graph.
how do you find volume of a cube
i really need help with my homework.
factorise 10x+25
Which of the following equations will have a negative solution?
A.m - 4 2/3 = 7 1/2
B.m - 71/2 = 4 2/3
C.7 1/2 + m = 4 2/3
D.-7 1/2 + m = 4 2/3
Since the value of m is -17/6, equation c has a negative solution.
What is the equation?An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.
It is given that,
7 1/2 + m = 4 2/3
We have to apply the arithmetic operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that are +, -, ×, and ÷.
15/2+m=14/3
Convert the fraction value and rearrange the equation as follows,
m=14/3-15/2
m=-17/6
Thus, the value of m is -17/6, equation c has a negative solution.
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Find the roots of the equation below. 7x2 + 3 = 8x
A) -4/7 (+ or -) (sqrt 5)/7
B) 4/7 (+ or -) (sqrt 5)/7
C) -4/7 (+ or -) i(sqrt 5)/7
D) 4/7 (+ or -) i(sqrt 5)/7
Answer:
x=4+-isqrt(5/7
Step-by-step explanation:
12 tenths plus 17 hundredths
Which graph represents the solution set for the system 2x + 5y ≤ 9 and 3x + 5y ≤ 9?
Answer:
The graph representing the solution is given below.
Step-by-step explanation:
We are given the system of inequality is,
[tex]2x + 5y\leq 9[/tex]
[tex]3x + 5y \leq 9[/tex]
Zero Test states that,
'After substituting the point (0,0) in the inequalities, if the result is true, then the solution region id towards the origin. If the result is false, the solution region is away from the origin'.
So, upon substituting (0,0) in the given inequalities, we get,
[tex]2x + 5y\leq 9[/tex] implies 0≤ 9, which is true.
[tex]3x + 5y \leq 9[/tex] implies 0≤ 9, which is true.
Thus, the solution region for both the inequalities is towards the origin.
Hence, upon plotting, the graph representing the solution set is given below.
A right triangle is removed from a rectangle to create the shaded region shown below. Find the area of the shaded region. Be sure to include the correct unit in your answer.
Answer:
The area of shaded region is 45 square units.
Step-by-step explanation:
The length of the rectangle is 8 and the breadth is 6.
Area of rectangle is
[tex]A_R=l\times b[/tex]
[tex]A_R=8\times 6=48[/tex]
The area of rectangle is 48 square units.
Since the opposite sides of a rectangle are equal, therefore the legs of the right angled triangle are
[tex]l_1=8-5=3[/tex]
[tex]l_2=6-4=2[/tex]
The area of right angled triangle is
[tex]A_T=\frac{1}{2}\times l_1\times l_2[/tex]
[tex]A_T=\frac{1}{2}\times 3\times 2=3[/tex]
The area of triangle is 3 square units.
The area of shaded region is
[tex]A=A_R-A_L=48-3=45[/tex]
Therefore the area of shaded region is 45 square units.