Answer:
[tex]y-2=\dfrac{1}{2}(x-6)[/tex]
Step-by-step explanation:
To find the equation of a line that passes through two given points, first find its slope by substituting the given points into the slope formula.
[tex]\boxed{\begin{minipage}{4.4cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\are two points on the line.\\\end{minipage}}[/tex]
Define the points:
(x₁, y₁) = (6, 2)(x₂, y₂) = (10, 4)Substitute the points into the slope formula:
[tex]\implies m=\dfrac{4-2}{10-6}=\dfrac{2}{4}=\dfrac{1}{2}[/tex]
Therefore, the slope of the line is ¹/₂.
[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}[/tex]
To find the equation in point-slope form, substitute the found slope and one of the given points into the point-slope formula:
[tex]\implies y-2=\dfrac{1}{2}(x-6)[/tex]
Write and solve an equation that represents the following math sentence.
Twenty-nine equals the sum of 11.5 and the quotient of a number and 2.5.
29 equals 11.5 plus m over 2.5; m = 43.75
29 equals 11.5 plus m over 2.5; m = 7
29 = 11.5 + 2.5m; m = 43.75
29 = 11.5 + 2.5m; m = 7
The algebraic expression which correctly represents the word phrase and it's solution are respectively; 29 equals 11.5 plus m over 2.5; m = 43.75.
What is the algebraic expression which correctly represents the given word phrase?It follows from the task content that the algebraic expression of which represents the word phrase is to be determined.
First, the quotient of a number, m and 2.5 can be represented algebraically as; m/2.5.
Therefore, since; Twenty-nine equals the sum of 11.5 and the quotient of a number and 2.5, we have;
29 = 11.5 + (m/2.5)
The solution is therefore as follows;
29 -11.5 = (m/2.5)
17.5 = (m/2.5)
m = 2.5 × 17.5
m = 43.75.
Therefore, the correct answer choice is; 29 equals 11.5 plus m over 2.5; m = 43.75.
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John took 30 minutes to bicycle to his grandmother’s house, a total of 2.5 kilometers. What was his speed in km/hr?
Answer:
5(km)/1(hr)
Step-by-step explanation:
You multiply the 30 minutes to make one hour, and with that, you also multiply the 2.5 kilometers to get 5. You multiply the 2.5 because we multiply both sides of an equation.
Uniform line movement.
We have that the data is:
Time (t) = 30 m = 0.5 h
Distance (d)= 2.5 km
Speed (v) = ?
Since it asks us for the speed in Km/h, we make a time conversion, from minutes to hours. Knowing that 1 hr = 60 minutes, then
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{30 \not{m}*\left(\frac{1 \ h}{60 \not{m})\right)=0.5 \ h } } \end{gathered}$}}[/tex]
To calculate the speed, divide the distance by the time. We apply the following formula:
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{Speed=\frac{Distance}{Time} } \end{gathered}$}}[/tex]
We substitute our data in the formula and solve:
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V=\frac{d}{t} } \end{gathered}$}}[/tex]
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V=\frac{2.5 \ km}{0.5 \ h} } \end{gathered}$}}[/tex]
[tex]\boxed{\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{V=5 \ km/h} \end{gathered}$} }}[/tex]
Juan's speed was: 5 km/h.A player hit a baseball from home plate 3 feet above the ground. The graph shows the height in feet of the baseball above the grounds as a quadratic function of x, the horizontal distance in feet of the baseball from home plate.What is the domain of this function?3≤x≤400≤y≤103≤y≤100≤x≤40
As the image attached shows, the horizontal distance reached by the ball is 40 feet.
Notice that the horizontal movement goes from 0 to 40, this means the domain is
[tex]0\leq x\leq40[/tex]Since x-values are domain values.
Therefore, the right answer is the last choice.A local video game store sells used games and new games. A new game costs$64, including tax. A used game costs $43, including tax. Luis bought 3 more used games than new games. Luis spent $343. How many used games did Luis purchase?
Given:
new game cost - $ 64
used game cost - $ 43
Luis spent $ 343
Required:
Number of used games Luis purchased
Solution
Let: x be the number of new games Luis bought
x + 3 be the number of used games Luis bought
Total cost = $ 343
Total Cost = (No. of new games bought)(Cost of new games) + (No. of used games bought)(Cost of used games)
$ 343 = ( x ) ( $ 64 ) + ( x + 3 ) ( $ 43 )
343 = 64x + 43 ( x + 3 )
343 = 64x + 43x + 129
343 - 129 = 107x
214 = 107x
2 = x
x = 2
x be the number of new games Luis bought 2
x + 3 be the number of used games Luis bought 2 + 3 = 5
Answer:
Luis purchased 5 used games
To check:
Substitute x into the equation,
343 = x ( 64 ) + (x + 3 ) (43)
343 = 2 ( 64 ) + (2 + 3)(43)
343 = 2 (64) + (5) (43)
343 = 128 + 215
343 = 343
The computed value of x satisfies the equation, Our answer is correct.
help pleaseeeeeeeeeeeeeeee
Answer:
(b) f(2) = 28
(c) f(-2) = -20
Step-by-step explanation:
f(2) = -2^3 + 7 x 2^2 - 2 x 2 + 12 f(-2) = -2^3 + 7 x -2^2 - 2 x -2 + 12
1. Calculate Exponents
-2^3 = -8 -2^3 = -8
2^2 = 4 -2^2 = -4
-8 + 7 x 4 - 2 x 2 + 12 -8 + 7 x -4 - 2 x -2 + 12
2. Multiply (left to right)
7 x 4 = 28 7 x -4 = -28
2 x 2 = 4 2 x -2 = -4
-8 + 28 - 4 + 12 -8 - 28 - (-4) + 12
3. Add (left to right)
-8 + 28 = 20 -8 + -28 = -36
20 - 4 = 16 -36 - (-4)= -32
16 + 12 = 28 -32 + 12 = -20
What is the area of the triangle 9 2 12
The given diagram is a traingle with base 12 units, height 4 units, and one side 9 units.
Since base (b) and height (h) are known, we can use the following formula for the area (A) of the triangle,
[tex]A=\frac{1}{2}bh[/tex]Substitute the values and simplify the expression,
[tex]\begin{gathered} A=\frac{1}{2}\times12\times4 \\ A=6\times4 \\ A=24 \end{gathered}[/tex]Thus, the area of the given triangle is 24 square units.
The balance of an account after t years can be found using the expression 4,500(1.75)^t where the initial balance was $4,500. By what percent does the account increase annually?
Solution
The Amount after t years is given by
[tex]A(t)=4500(1.75)^t[/tex]The Percentage annual increase is
[tex]\begin{gathered} \frac{A(1)-A(0)}{A(0)}\times100\% \\ \Rightarrow\frac{4500(1.75)-4500}{4500}\times100\% \\ \Rightarrow0.75\times100\%=75\% \end{gathered}[/tex]Hence, the account increase 75% annually
Please help! I can also change the vent graph to different kinds as well.
Answer:
See below and attached Venn Diagram3
Step-by-step explanation:
No need to change Venn diagrams It is perfectly clear
We will represent the different sets of people using letters A and B for convenienceThat total number of people who "Had Drowsiness" is represented by the maroon circle to the left. )Note some of those participants also belong to the category had Nausea)We will represent the set of all people who had drowsiness as A and the number of people in that set as n(A) = 95 (given)Similarly, the total number of people who had nausea is represented by the aquamarine circle to the right. (Note that some of these people also had drowsiness) We will represent the set of all people who had nausea as B and the number of people in that set as n(B) = 77 (given)The number of people who had both drowsiness and nausea is the intersection area of the two i.e. the Venn diagram area in the middle which overlaps both the circles. This is represented as A and B and we are given that this number n(A and B) = 58The completed Venn Diagram is attached for your convenience
Vector A and B are at right angles and have the same initial point. Find the magnitude and direction from vector A of the resultant vector.
We need to find the magnitude and direction from vector A of the resultanr vector.
How many gallons of the water are in the filled tank? You must show work with dimensional analysis using the above fact.
From the question, we can deduce the following:
Length = 5.6 meters
Width = 7.3 meters
Height = 4.5 meters
Where:
1 ft = 0.3048 m
1 cubic foot = 7.48 gallons
Let's find the amount of gallons of water in the tank if the tank is filled to 1 foot below the top of the tank.
Let's first sketch the tank.
We have:
Now, let's find the volume of the tank(total amount of water it can carry).
Apply the formula:
[tex]V=l\times w\times h[/tex]Thus, we have:
[tex]\begin{gathered} V=5.6\times7.3\times4.5 \\ \\ V=183.96m^3 \end{gathered}[/tex]Now, to find the amount of water presently in the tank since it is filled 1 ft below the top of the tank, we have:
[tex]\begin{gathered} V=5.6\times7.3\times(4.5-0.3048) \\ \\ V=5.6\times7.3\times4.1952 \\ \\ V=171.5m^3 \end{gathered}[/tex]Therefore, the volume of the water in the tank is 171.5 cubic meters.
Now, let'c convert from cubic meters to gallons.
Where:
1 foot = 0.3048 m
[tex]1meter=\frac{1}{0.3048}=3.28\text{ feet}[/tex]Thus, we have:
[tex]\begin{gathered} 1\text{ cubic meter = 3.28}^3=35.29\text{ cubic f}eet \\ \\ 171.5\text{ cubic meters = 171.5 x 35.29 = }6052.235\text{ cubic fe}et \end{gathered}[/tex]Now, to convert from cubic feet to gallons, where:
1 cubci foot = 7.48 gallon
We have:
[tex]6052.235\times7.48=45273.86\text{ gallons}[/tex]Therefore, there are 45273.86 gallons of water filled in the tank.
If the Manufacturing Overhead account is closed proportionally to Work in Process, Finished Goods, and Cost of Goods Sold, the related entry will include a ________.
If the Manufacturing Overhead account is closed proportionally to Work in Process, Finished Goods, and Cost of Goods Sold, the related entry will include a Credit to cost of goods sold for $12000
How to solve for the costThe cost of a good is the total amount that was used in the purchase of a particular good form the market.
we have the manufacturing overhead to be = 30000 dollars
the work in progress = 30000 x 25 %
= 30000 x 0.25
= 7500
The finished goods = 30000 x 35 %
= 30000 x 0.35
= 10500
The cost of good sold = 30000 - 10500 - 7500
= 12000
Hence the cost of good sold is 12000
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complete questionManufacturing overhead applied $ 150,000
Actual amount of manufacturing overhead costs 120,000
Amount of overhead applied during the year that is in:
Work in Process $ 37,500 25 %
Finished Goods 52,500 35 %
Cost of Goods Sold 60,000 40 %
Total overhead applied $ 150,000 100 %
If the Manufacturing Overhead account is closed proportionally to Work in Process, Finished Goods, and Cost of Goods Sold, the related entry will include a ________.
debit to Cost of Goods Sold for $12,000
credit to Cost of Goods Sold for $12,000
credit to Cost of Goods Sold for $30,000
debit to Work in Process for $7,500
15. The life of a 9-volt battery is normally distributed with a mean of = 2000 hours and a standarddeviation of = 40 hours. What is the proportionof batteries with an average life of 2100 hours ormore? (enter the answer as a percent rounded tothe nearest hundredth as needed)
To answer this question we will use the following formula for the z-score:
[tex]\begin{gathered} z=\frac{x-\mu}{\sigma}, \\ where\text{ }x\text{ is the observed value, }\mu\text{ is the mean, and }\sigma\text{ is the standard deviation.} \end{gathered}[/tex]Substituting = 2000h, = 40h and x=2100h we get:
[tex]z=\frac{2100h-2000h}{40h}.[/tex]Simplifying the above result we get:
[tex]z=\frac{100h}{40h}=2.5.[/tex]Using tables we get:
[tex]P(z<2.5)=0.99379[/tex]Then:
[tex]P(z\geq2.5)=1-0.99379=0.00621.[/tex]The above result as a percentage is:
[tex]0.621\%.[/tex]Answer:
[tex]0.62\%.[/tex]
7Lines a and bare parallel cut by transversal line t solve for the value of x25x + 4a3x + 14
These angles measure the same they are interior alternate angles.
3x + 14 = 5x + 4
Solve for x
3x - 5x = 4 - 14
Simplify like terms
-2x = -10
x = -10/-2
Result
x = 5
you pick a marble and flip a coin how many outcomes are possible
Solution
A coin has two possible outcome
We have 5 marbles
[tex]2\times5=10[/tex]The final answer
[tex]10[/tex]The wholesale price for a chair is 194$ . A certain furniture store marks up the wholesale price by 35%. Find the price of the chair in the furniture store.
The price of the chair will be 261.9 $ .
One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entirety and 200 percent specifies twice the given quantity. For example, 1 percent of 1,000 chickens equals 1/100 of 1,000, or 10 chickens; 20 percent of the quantity is 20/100 1,000, or 200.
If we say, 5%, then it is equal to 5/100 = 0.05.
To solve percent problems, you can use the equation, Percent · Base = Amount, and solve for the unknown numbers. Or, you can set up the proportion, Percent = , where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.
Based on given conditions formulate
x = 194 ×(35%+1)
x= 194 ×1.35
x = 261.9 $ .
Thus The price of the chair will be 261.9 $ .
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Andre was trying to write 7^4/7^-3 with a single exponent and write 7^4/7^-3= 7^4-3=7^1 Exploit to Andre what his mistake was and what the answer should be...PLEASE THE ANSWER IS URGENT!!
Here, we want to get what Andre's mistake was and correct it
To answer this, we are supposed to use the division law of indices
We have this as;
[tex]\frac{a^x}{a^y\text{ }}=a^{x-y}[/tex]Now, in the case of this question, x is 4 and y is -3
So, we have the expression as;
[tex]7^{4-(-3)}=7^{4+3}=7^7[/tex]His mistake is thus adding the exponents instead of subtracting
PLS HELP ME ASAP!!!!
The middle value in a sorted, ascending or descending list of numbers is known as the median, and it has the potential to describe a data collection more accurately than the average does.
What is meant by data set?A data set is a group of figures or values related to a specific topic. a list of words, numbers, or symbols, the values of which are frequently the outcomes of an investigation or observation. There are typically multiple variables in data sets.
A dataset is an organized group of data typically connected to a certain body of work. An organized collection of data kept as several datasets is called a database.
In an ordered data set, the median is the value in the middle. The mean is calculated by dividing the sum of all values by the total number of values. The middle value in a sorted, ascending or descending list of numbers is known as the median, and it has the potential to describe a data collection more accurately than the average does.
Therefore, the correct answer is option c) The northbound cars were generally faster because the median for the northbound cars is greater than the median for the southbound cars.
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Use the rules of significant figures to answer the following question:43.5694 * 22.07A. 961.58B. 961C. 961.577D. 961.7
we have that
43.5694 * 22.07=961.576658
therefore
the answer is
961.577 -----> 6 figures
(remember that 43.5694 has 6 figures)
option C
Describe a situation that can be modeled using the given simulation. 1 marble chosen from a bag containing 11 red marbles and 4 blue marbles a. Determine the probability of selecting a month from January through November and the month begins with the letter J. b. Determine the probability of surveying 11 people and 4 of them like playing checkers. c. Determine the probability of selecting a pair of tennis shoes from a shoe collection that contains 4 dress shoes and 7 tennis shoes. d. Determine the probability of selecting an apple from a basket that contains 11 apples and 4 oranges.
Answer:
Explanation:
A bag contains 11 red marbles and 4 blue marbles.
• The total number of marbles = 11+4=15
1 marble of any color is chosen random from the bag with a probability of 1/15.
A situation that can be modeled using such a situation must be one that also has a probability of 1/15.
In Option D,
Given: Ray BD bisects prove: <1 and <3 are supplementary
Let's begin by listing out the information given to us:
[tex]\begin{gathered} |BD|\text{ bisects }|EBC|\text{ into equal parts}\Rightarrow\angle1=\angle2 \\ \angle2\text{ is supplementary with }\angle3;\angle2+\angle3=180^{\circ} \\ \therefore\angle1+\angle3=180^{\circ} \end{gathered}[/tex]Find the value of f(4) for the func
f(x) = -3(x + 2)
f(4) = (Simplify your answer.)
Answer: f(4) = -18
Step-by-step explanation:
f(4) = -3(4 + 2)
= -3(6)
= -18
A baseball team has home games on Wednesday and Saturday. The two games together earn $4920.00 for the team. Wednesday's game generates $880.00 less than Saturday's game. How much money did Wednesday's game generate?
Wednesday's game generated $2020
Explanations:Let the amount generated by Wednesday's game be x
Let the amount generated by Saturday's game be y
The two games together earn $4920 for the team. That is,
x + y = 4920....................(1)
Wednesday's game generates $880 less than Saturday's game
x = y - 880.................(2)
Make y the subject of the formula in equation (2)
y = x + 880..................(3)
Substitute equation (3) into equation (2)
x + x + 880 = 4920
x + x = 4920 - 880
2x = 4040
Divide both sides by 2
2x/2 = 4040/2
x = 2020
Wednesday's game generated $2020
The sum of the squares of three consecutive odd numbers is 83. Find the numbers.
In order to represent three consecutive odd numbers, we can use the expressions "x", "x+2" and "x+4".
If we add the square of each number, the result is 83, so we can write the following inequality:
[tex]\begin{gathered} x^2+(x+2)^2+(x+4)^2=83\\ \\ x^2+x^2+4x+4+x^2+8x+16=83\\ \\ 3x^2+12x+20=83\\ \\ 3x^2+12-63=0\\ \\ x^2+4x-21=0 \end{gathered}[/tex]Let's solve this quadratic equation using the quadratic formula, with a = 1, b = 4 and c = -21:
[tex]\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4a}c}{2a}\\ \\ x=\frac{-4\pm\sqrt{16+84}}{2}\\ \\ x=\frac{-4\pm10}{2}\\ \\ x_1=\frac{-4+10}{2}=\frac{6}{2}=3\\ \\ x_2=\frac{-4-10}{2}=\frac{-14}{2}=-7 \end{gathered}[/tex]If we assume the numbers are positive, the numbers are 3, 5 and 7.
(The other result, with negative numbers, would be -7, -5 and -3).
(1 point) The expression 64g4y2 64gʻy2 equals kg"ys where r, the exponent of g, is: and s, the exponent of y, is: and k, the leading coefficient is:
We are asked to find the final expression in exponent for, of the product of two radical expressions as shown below:
We will be addressing each factor at a time, first in the cubic root, and second in the square root.
Cubic root:
notice that the pure number (64) can be written as the perfect cube of the number 4 since 4^3 = 64. That means that a factor 4 will come out of the cubic root, and no factor 4 will be left in the root.
Now we are going to address the variables g and y which are inside the root, and try to find if they contain any perfect cube expression (perfect cubes will cancel out of the cubic root simplifying then the notation).
We notic that g^4 can be written as g^3 times g, so g^3 will come out of the cubic root as "g", and a factor "g" would be left inside the cubic root. Similarly, the factor y^2 cannot get out of the cubic root because it doesn't have a perfect cube in it.
We are then left with the following expression:
[tex]4g\sqrt[3]{g\cdot y^2}[/tex]Now, we recall the property of fractional exponents associated with roots of a factor:
[tex]\sqrt[n]{x}=x^{\frac{1}{n}}[/tex]Using this, the cubic root can be expressed as fractional exponents, as we show below:
[tex]4g\sqrt[3]{g\cdot y^2}=4g\cdot g^{\frac{1}{3}}\cdot y^{\frac{2}{3}}=4g^{\frac{4}{3}}\cdot y^{\frac{2}{3}}[/tex]Now we proceed to simplify the expression with the square root. This is much simpler, since all factor inside can be written as perfect squares:
64 = 8^2
g^4 = (g^2)^2 (the double square of g)
y^2 is already a perfect square.
Therefore, the square root is simplified entirely and we are left with the following factors:
8 g^2 y
Now we combine via the multiplication what we found for the cubic root and what we found for the square root:
[tex](4\cdot g^{\frac{4}{3}}\cdot y^{\frac{2}{3}})\cdot(8\cdot g^2\cdot y)=32\cdot g^{\frac{10}{3}}\cdot y^{\frac{5}{3}}[/tex]Therefore the power (r) of the factor g is the fraction: 10/3
The power (s) of the factor y is the fraction 5/3
and the constant number "k" is 32.
M={z / z is an integer and -2
The given set in roster method is written as {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}.
Given that, M={x : x is an integer and 2 ≤ x ≤20.
What is roster method?The roster method is defined as a way to show the elements of a set by listing the elements inside of brackets.
Now, the given set in roster method can be written as follows:
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
Therefore, the given set in roster method is written as {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}.
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"Your question is incomplete, probably the complete question/missing part is:"
Write the given set in roster method: M={x : x is an integer and 2 ≤ x ≤20
use basic trigonometric identities to simplify the expression: sin^3 (x) + cos^2 (x) sin (x)
Given the following trigonometric expression:
[tex]sin^3x+cos^2x*sinx=?[/tex]We will simplify the expression as follows:
First, taking sin (x) as a common
[tex]=sin\text{ }x*(sin^2x+cos^2x)[/tex]Second, we will use the following trigonometric identity:
[tex]sin^2x+cos^2x=1\rightarrow(1)[/tex]Finally, substitute the trigonometric identity number (1) into the given expression
[tex]=sin\text{ }x*1=sin\text{ }x[/tex]So, the answer will be sin (x)
Would the inverse of this graph be a function? Why or why not?
We will have the following:
*First: We would determine the piecewise function that describes the graph.
Since we can see that the function increases by a factor of 2 and decreases by the same factor we would have:
[tex]f(x)=\begin{cases}2x\colon x\ge0\land x\le4 \\ \\ -2x+16\colon x>4\land\le8\end{cases}[/tex]This is:
*Second: We calculate the inverse of the piecewise function:
[tex]f^{-1}(x)=\begin{cases}\frac{x}{2}\colon x\ge0\land x\le4 \\ \\ -\frac{x-16}{2}\colon x>4\land x\le8\end{cases}[/tex]That is:
From this we can see that the inverse of that graph would in fact represent a function, a non-continuous function. [It will represent a function as long as it follows the parameters stablished in the first point]
4/5 × (12÷2+4)-1 what is the value of the expression
Given the expression
[tex]\frac{4}{5}\cdot(12\div2+4)-1[/tex]To find the value of the expression you have to follow the order of operations:
- Parentheses
- Exponents
- Multiplication/Division
- Addition/ Subtraction
• First, you have to solve the operations inside the parentheses term:
[tex]12\div2+4[/tex]Solve the division first and then the sum:
[tex]12\div2+4=6+4=10[/tex]• Write the result on the original expression:
[tex]\begin{gathered} \frac{4}{5}\cdot(12\div2+4)-1 \\ \frac{4}{5}\cdot10-1 \end{gathered}[/tex]• The next step is to solve the multiplication and finally solve the difference
[tex]\frac{4}{5}\cdot10-1=8-1=7[/tex]The value of the expression is 7
4. Answer the following questions based on the information below:Han wants to build a dog house. He makes a list of the materials needed:• At least 60 square feet of plywood for the surfaces• At least 36 feet of wood planks for the frame of the dog house• Between 1 and 2 quarts of paintHan's budget is $65. Plywood costs $0.70 per square foot, planks of wood cost $0.10 per foot, andpaint costs $8 per quart.
Part A
The variable are;
Plywood, wood planks, quarts of paint
Part B
Let p represent Plywood
Let w represent Plank of wood
[tex]1Part c
Let p represent Plywood
Let w represent Plank of wood
Let q represent quarts of paint
[tex]0.7p+0.1w+8q\leq65[/tex]Find all the values of x where the tangent line is horizontal.3f(x) = x³ - 4x² - 7x + 12X=(Use a comma to separate answers as needed. Type an exact answer, using radicals
Given the function:
[tex]h(x)=x^3-4x^2-7x+12[/tex]Find the first derivative:
[tex]h^{\prime}(x)=3x^2-8x-7[/tex]The first derivative gives us the slope of the tangent line to the graph of the function. When the tangent line is horizontal, the slope is 0, thus:
[tex]3x^2-8x-7=0[/tex]This is a quadratic equation with coefficients a = 3, b = -8, c = -7.
To calculate the solutions to the equation, we use the quadratic solver formula:
[tex]$$x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}$$ [/tex]Substituting:
[tex]x=\frac{-(-8)\pm\sqrt{(-8)^2-4(3)(-7)}}{2(3)}[/tex]Operate:
[tex]\begin{gathered} x=\frac{8\pm\sqrt{64+84}}{6} \\ \\ x=\frac{8\pm\sqrt{148}}{6} \end{gathered}[/tex]Since:
[tex]148=2^2\cdot37[/tex]We have:
[tex]\begin{gathered} x=\frac{8\pm2\sqrt{37}}{6} \\ \\ \text{ Simplifying by 2:} \\ \\ x=\frac{4\pm\sqrt{37}}{3} \end{gathered}[/tex]There are two solutions:
[tex]\begin{gathered} x_1=\frac{4+\sqrt{37}}{3} \\ \\ x_2=\frac{4-\sqrt{37}}{3} \end{gathered}[/tex]