Consider the function f(x) = 9x - x^2Compute the derivative values of the following:f’ (0) =f'(1) =f'(2) =f'(3)Conjecture a formula for f(a) that depends only on the value a. That is, in the same way that we have a formula for f(x) (recall f(x) = 9x - x^2), see if you can use your work above to guess a formula for f'(a) in terms
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Solve for f'(0), f'(1), f'(2), f'(3)
[tex]\begin{gathered} f^{\prime}(x)=9-2x \\ f^{\prime}(0)=9-2(0) \\ f^{\prime}(0)=9-0 \\ f^{\prime}(0)=9 \\ \\ f^{\prime}(1)=9-2(1) \\ f^{\prime}(1)=9-2 \\ f^{\prime}(1)=7 \\ \\ f^{\prime}(2)=9-2(2) \\ f^{\prime}(2)=9-4 \\ f^{\prime}(2)=5 \\ \\ f^{\prime}(3)=9-2(3) \\ f^{\prime}(3)=9-6 \\ f^{\prime}(3)=3 \end{gathered}[/tex]the conjecture for f'(a)
[tex]\begin{gathered} \text{Since }f^{\prime}(x)=9-2x,\text{ then} \\ f^{\prime}(a)=9-2a \end{gathered}[/tex]Related Questions
For a conditional statement, the contrapositive of the converse is always logicallyequivalent to the inverse.
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Yes, it is True
If the converse is true, then the inverse is also logically true.
Find the surface area of the cylinder to the nereast tenth of a square unit. Use 3.14 for PI.
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Given:
Height of the cylinder is, h = 18.2 cm.
Radius of the cylinder is, r = 3 cm.
The objective is to find the surface area of the cylinder.
The formula to find the surface area of the cylinder is,
[tex]SA=2\pi r^2+2\pi rh[/tex]Now substitute the given values in the above equation.
[tex]\begin{gathered} SA=(2\cdot3.14\cdot3^2)+(2\cdot3.14\cdot3\cdot18.2) \\ =(2\cdot3.14\cdot9)+(2\cdot3.14\cdot3\cdot18.2) \\ =56.5+342.9 \\ =399.4cm^2 \end{gathered}[/tex]Hence, option (B) is the correct answer.
What is the least common multiple of 62 +39 - 21 and 6x² +54x+84? O 6x² +54x+84 6x² +93x +63 62³ +52x² + 111x − 42 O 12x³ + 102x² + 114 - 84
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SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given expressions
[tex]6x^2+39x-21\text{ }and\text{ }6x^2+54x+84[/tex]STEP 2: Define the least common multiple
The Least Common Multiple ( LCM ) is also referred to as the Lowest Common Multiple ( LCM ) and Least Common Divisor ( LCD) . For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b.
STEP 3: Find the LCM
Factorize the first expression
[tex]6x^2+39x-21=3(2x-1)(x-7)[/tex]Factorize the second expression:
[tex]6x^2+54x+84=3(x+2)(x+7)[/tex]Calculating the LCM, we have:
[tex]\begin{gathered} \mathrm{Multiply\:each\:factor\:with\:the\:highest\:power:} \\ 2\cdot \left(2x-1\right)\cdot \:3\cdot \left(x+2\right)\cdot \left(x+7\right) \\ Simplify \\ 6\left(2x-1\right)\left(x+2\right)\left(x+7\right) \end{gathered}[/tex]Evlauating the result gives:
[tex]12x^3+102x^2+114x-84[/tex]Hence, the LCM is:
[tex]12x^3+102x^2+114x-84[/tex]In Fig. 12.8, PQRS is a parallelogram, HSR isa straight line and HPQ = 90°. If|HQ| = 10 cm and |PQ| = 6 cm, what is thearea of the parallelogram?
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Answer
48 cm²
Step-by-step explanation
First, we need to calculate the height of the parallelogram, segment HP. Applying the Pythagorean theorem to triangle HPQ, we get:
[tex]\begin{gathered} HQ^2=HP^2+PQ^2 \\ \text{ Substituting with HQ = 10 cm, and PQ = 6 cm, and solving for HP:} \\ 10^2=HP^2+6^2 \\ 100=HP^2+36 \\ 100-36=HP^2 \\ 64=HP^2 \\ \sqrt{64}=HP \\ HP=8\text{ cm} \end{gathered}[/tex]The area of a parallelogram is calculated as follows:
[tex]A=base\times height[/tex]In this case, the height is HP = 8 cm, and the base is PQ = 6 cm. Then the area of parallelogram PQRS is:
[tex]\begin{gathered} A=HP\times PQ \\ A=8\times6 \\ A=48\text{ cm}^2 \end{gathered}[/tex]
For csc 330:a) state value of the ratio exactlyb) find one equivalent expressionc) draw a diagram to illustarte.
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For csc 330:
a) state value of the ratio exactly
b) find one equivalent expression
c) draw a diagram to illustarte.
SolutionPart A[tex]-2[/tex]Part bRecall:
[tex]csc=\frac{1}{sin330}[/tex]Part cFind a dishes between an order pairs (-2 -1) and (-5,-4) round your solution to the nearest 10th, if necessary.
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To solve this problem, we will use the following formula for the distance between two points (x₁,y₁) and (x₂,y₂):
[tex]d=\sqrt[]{(x_1-x_2)^2+(y_1-y_2)^2}.[/tex]Substituting the given points in the above formula, we get:
[tex]d=\sqrt[]{(-2-(-5))^2+(-1-(-4))^2}.[/tex]Simplifying the above result, we get:
[tex]d=\sqrt[]{(-2+5)^2+(-1+4)^2}=\sqrt[]{3^2+3^2}=\sqrt[]{18}.[/tex]Therefore:
[tex]d\approx4.2.[/tex]Answer:
[tex]4.2[/tex]Question 9 (3 points)y = -x3 + 2x + 3d: {-2, 1,2} r: {3Blank 1:Blank 2:IBlank 3:
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Question #6 6) Find the hypotenuse of a right triangle if side a is 3 feet and side b is 4 feet
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Side a : 3
side b : 4
Since it's a right triangle we can apply the Pythagorean theorem:
c^2 = a^2+b^2
Where c is the hypotenuse.
Replacing with the values given and solving for c:
c^2 = 3^2+4^2
c^2 = 9 + 16
c^2 = 25
c =√25
c = 5
Answer : 5 feet
Sketch the asymptotes and the graph of the function. Identify the domain and range. y = 7/x+4 -7Choose the correct graph below (the asymptotes are shown as red dashed lines).
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In the horizontal asymptotes:
If the exponent of the numerator is the same as the exponent of the denominator, then:
[tex]\begin{gathered} y=\text{ }\frac{a}{b} \\ y=\text{ }\frac{7}{1} \\ y=7 \end{gathered}[/tex]But, when there's a translation, then y= k, which means y= - 7.
In the vertical asymptotes:
You have to equal the denominator to 0, so:
[tex]\begin{gathered} x+\text{ 4= 0 } \\ x=\text{ -4} \end{gathered}[/tex]The domain is all real numbers except -4 and the range is all real numbers except -7.
Now we have to check the graphs.
The correct graph is C. When x is -4 and y is -7.
Suppose that sec(t) = 3/2 and that t is in quadrant IV. Find the exact value of tan(t).
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Using the trigonometric identities
[tex]\sec ^2(t)=\tan ^2(t)+1[/tex]We want to find out the tan(t), then we can manipulate that formula and find tan(t) in function of sec(t).
[tex]\tan ^2(t)=\sec ^2(t)-1[/tex]Now we can do square roots on both sides
[tex]\tan (t)=\pm\sqrt[]{\sec ^2(t)-1}[/tex]We know that sec(t) = 3/2, then let's put it in our formula and simplify
[tex]\begin{gathered} \tan (t)=\pm\sqrt[]{\frac{3^2}{2^2}-1} \\ \\ \tan (t)=\pm\sqrt[]{\frac{9}{4}-1} \\ \\ \tan (t)=\pm\sqrt[]{\frac{9}{4}-\frac{4}{4}} \\ \\ \tan (t)=\pm\sqrt[]{\frac{5}{4}} \end{gathered}[/tex]We can simplify and remove 4 from the square root, and we have
[tex]\tan (t)=\pm\frac{\sqrt[]{5}}{2}[/tex]But which value is correct? the positive or the negative? Now we must use the information that the problem tells us, it says that t is in the quadrant IV, the tangent in quadrant IV is negative, then
[tex]\tan (t)=-\frac{\sqrt[]{5}}{2}[/tex]A sphere and a cylinder each have the same radius. The cylinder has a height that is triple the radius. Which figure has a greater volume, the cylinder or the sphere?
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we calculate the volume of the two solids
Sphere
[tex]V_s=\frac{4}{3}\times\pi\times r^3[/tex]Cylinder
[tex]\begin{gathered} V_c=\pi\times r^2\times h \\ V_c=\pi\times r^2\times3r \\ V_c=\pi\times3r^3 \end{gathered}[/tex][tex]\begin{gathered} \frac{4}{3}\times\pi\times r^3 \\ \\ 3\times\pi\times r^3 \end{gathered}[/tex]the comparison allows us to see that the volume of the cylinder is greater than that of the sphere
f(x) = 5x2 + 2x − 3What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph.
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EXPLANATION:
Given;
We are given a quadratic function as shown below;
[tex]f(x)=5x^2+2x-3[/tex]Required;
We are required to show the steps required to graph this function.
Step-by-step solution;
To do this, we would need to set up a table of values for x and y.
We shall take input values that is x, and use this to get corresponding output value, that is y.
[tex]\begin{gathered} When\text{ }x=0: \\ y=5(0)^2+2(0)-3 \end{gathered}[/tex][tex]y=0+0-3=-3[/tex]Therefore, we have the coordinates (0, -3)
[tex]\begin{gathered} When\text{ }x=1: \\ y=5(1)^2+2(1)-3 \end{gathered}[/tex][tex]y=5+2-3=4[/tex]This gives us the coordinates, (1, 4)
[tex]\begin{gathered} When\text{ }x=2: \\ y=5(2)^2+2(2)-3 \end{gathered}[/tex][tex]y=20+4-3=18[/tex]This too gives us the coordinates, (2, 18).
Note that the same can be done for the left side of the graph (where the x values are negative).
Hence, using the same procedure as shown above, we would have;
[tex]\begin{gathered} When\text{ }x=-1,y=0 \\ That\text{ }is,\text{ }(-1,0) \end{gathered}[/tex][tex]\begin{gathered} When\text{ }x=-2,y=13 \\ That\text{ }is,\text{ }(-2,13) \end{gathered}[/tex][tex]\begin{gathered} When\text{ }x=-3,y=36 \\ That\text{ }is,\text{ }(-3,36) \end{gathered}[/tex]We can go on and plot as many points as the graph page can accommodate using this same procedure.
Next step is to connect the points as shown by the coordinates.
Using a graphing calculator, the graph of the quadratic function given will be as shown below;
The function f(x) = x³ - 3x² + 2x rises as x grows very large. O A. True O B. False
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We have to find what happens with f(x) when x grows very large.
The function is:
[tex]f(x)=x^3-3x^2+2x[/tex]When x grows very large, the term that has a higher degree will be the one that have the greatest effect in the value of f(x).
In this function, the term with the highest degree is x³.
As x grows very large, so does x³. Then, we can conclude that f(x) will also rise as does x.
Answer: True.
Expand each expression. Use your calculator to check that both forms are equivalent.
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The Solution.
we shall expand each of the given expressions:
a.
[tex]\begin{gathered} (x-5)^2 \\ =(x-5)(x-5) \\ =x(x-5)-5(x-5) \\ =x^2-5x-5x+25 \\ =x^2-10x+25 \end{gathered}[/tex]b.
[tex]\begin{gathered} (x-7)^2 \\ =(x-7)(x-7) \\ =x(x-7)-7(x-7) \\ =x^2-7x-7x+49 \\ =x^2-14x+49 \end{gathered}[/tex]c.
[tex]\begin{gathered} (x-2)^2 \\ =(x-2)(x-2) \\ =x(x-2)-2(x-2) \\ =x^2-2x-2x+4 \\ =x^2-4x+4 \end{gathered}[/tex]Let f(x)=x^3 + 2x^2 - 18 . For what values of x is f(x) = 9? Enter your answers as a comma-separated list
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Answer
The values of x for which f(x) = 9 include
x = 2.46035
x = -2.23018 + 2.44956i
x = -2.23018 - 2.44956i
In comma-seperated list
x = [3.83543, (-2.23018 + 2.44956i), (-2.23018 - 2.44956i)]
Explanation
We are told that
f(x) = x³ + 2x² - 18
We are then asked to find the values of x for which f(x) = 9
When f(x) = 9,
9 = x³ + 2x² - 18
x³ + 2x² - 18 = 9
x³ + 2x² - 18 - 9 = 0
x³ + 2x² - 27 = 0
Now, we have to solve this polynomial
On inputing this on a clculator, the 3 answers for this polynomial expression include
x = 2.46035
x = -2.23018 + 2.44956i
x = -2.23018 - 2.44956i
Hope this Helps!!!
Which expression best estimates 6O 7-2○ 6-18-3○ 3-2NEATRY10:DAITEELKATEER EEN ANDER HEAADEMPIRENTE] PAR MEpar spenna#] [3]5\y\£x{] =>DEEEN TOEGERHENDEROgeJENTERE3위 - 10
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Step 1
Given;
[tex]6\frac{3}{4}\div1\frac{2}{3}[/tex]Required; To find Which expression best estimates the expression.
Step 2
[tex]\begin{gathered} 6\frac{3}{4}\text{ =6.75}\approx7 \\ 1\frac{2}{3}\approx2 \end{gathered}[/tex]Thus the answer will be;
[tex]7\div2[/tex]Solving rational equationsWhich step is the error.SOLVE FOR APlease explain step by step
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To find the error, we repeat the steps correctly:
i)
[tex]b=\frac{1}{2}\cdot\sqrt[]{a-1}[/tex]ii) We pass the 2 dividing on the right, multiplicating at the left:
[tex]2b=\cdot\sqrt[]{a-1}[/tex]iii) We take the square on both sides:
[tex]\begin{gathered} (2b)^2=(\sqrt[]{a-1})^2 \\ 2^2b^2=a-1 \\ 4b^2=a-1 \end{gathered}[/tex]iv) Finally we pass the -1 on the right as +1 to the left:
[tex]4b^2+1=a[/tex]Answer
Comparing the steps, we see that the error was in step III, the square of 2b is 4b², not 2b².
ABC is a triangle for which < ABC = 90°, AB= 10cm and BC=15cm, Find Ac? how do I solve this?
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The Pythagorean theorem states:
[tex]c^2=a^2+b^2[/tex]where a and b are the legs and c is the hypotenuse of a right triangle.
Applying this theorem to triangle ABC (see the above diagram):
[tex]AC^2=AB^2+BC^2[/tex]Substituting with AB = 10 cm, BC = 15 cm, and solving for AC:
[tex]\begin{gathered} AC^2=10^2+15^2 \\ AC^2=100+225 \\ AC^2=325 \\ AC=\sqrt[]{325} \\ AC\approx18\operatorname{cm} \end{gathered}[/tex]What is the solution to the following equation? 3(X - 4)- 5 = X - 3 O A. X = 12 B. X = 8 O C. x=7 O D. *= 3
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The initial equation is:
[tex]3(x-4)-5=x-3[/tex]Now we can distribute the 3 into the parenthesis so:
[tex]3x-12-5=x-3[/tex]Now we move all variable to the left of the equation and the constants to the right so:
[tex]3x-x=-3+12+5[/tex]Finally we simplify so:
[tex]\begin{gathered} 2x=14 \\ x=\frac{14}{2} \\ x=7 \end{gathered}[/tex]So the answer is C) x = 7
In order to solve 7x² + 5x - 3 = 0 by the quadratic formula
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We can solve a quadratic equation of the form:
[tex]ax^2+bx+c=0[/tex]The quadratic formula:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]Given the equation:
[tex]\begin{gathered} 7x^2+5x-3=0 \\ a=7,b=5,c=-3 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} x=\frac{-5\pm\sqrt{5^2-4\times7\times-3}}{2\times7} \\ =\frac{-5\pm\sqrt{25-(-84)}}{14} \\ =\frac{-5\pm\sqrt{25+84}}{14} \\ =\frac{-5\pm\sqrt{109}}{14} \\ \text{Therefore:} \\ x=\frac{-5+\sqrt{109}}{14}\lor x=\frac{-5-\sqrt{109}}{14} \\ x=0.3886\lor x=-1.1029 \end{gathered}[/tex]What’s the sun of the interior angle measures for convex nonagon.
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We need to find the sum of the interior angles of a convex nonagon.
First, let's remember that a nonagon is a polygon with 9 sides.
Also, the formula to find the sum of the interior angles for a convex polygon of n sides is:
[tex]\text{ sum of interior angles }=(n-2)\cdot180\degree[/tex]So, in this case, we have n = 9. Using this value in the above formula, we obtain:
[tex]\begin{gathered} \text{ sum of interior angles of a convex nonagon }=(9-2)\cdot180\degree \\ \\ =7\cdot180\degree \\ \\ =1260\degree \end{gathered}[/tex]Therefore, the answer is 1260º.
Find the value of 9y+1 given that -2y-1=5.Simplify your answer as much as possible.
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Answer
The value of 9y + 1 is -26
Explanation:
Given the below equation
-2y - 1 = 5
Step 1: find y
To find y, firstly collect the like terms
-2y = 5 + 1
-2y = 6
Divide both sides by -2
-2y / -2 = 6/-2
y = -3
Given 9y + 1
Substitute the value of y = -3
9(-3) + 1
= -27 + 1
= - 26
Therefore, the value of 9y + 1 is -26
9√x +7√x simplify the expression
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We will have the following:
[tex]9\sqrt{x}+7\sqrt{x}=16\sqrt{x}[/tex]The population of mice doubles every year. There are 10 mice in the population. Write an exponential function to model this situation:Determine the population of the mice in 11 years. How many years will it take for the population of mice to reach at least 10 million?
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Let's list down the given information in the problem.
1. initial value = 10 mice
2. rate = doubles every year = 200%
3. time = 11 years
4. final value = at least `10 million
The equation of an exponential function goes by this pattern:
[tex]f(x)=ab^x[/tex]where a = initial value, b = growth rate, x = time in years
From the given information, we can write an exponential model of the situation by plugging in those given data (1 and 2 only) to the pattern above.
[tex]f(x)=10(2)^x[/tex]The exponential model is f(x) = 10(2)^x as shown above.
After 11 years, the population will be: (plug in x = 11 to the model)
[tex]\begin{gathered} f(x)=10(2)^{11} \\ f(x)=10(2048) \\ f(x)=20,480 \end{gathered}[/tex]After 11 years, the population of the mice will have been 20, 480.
To calculate how many years it will take the population to reach at least 10 million, we will have to assume that f(x) = 10 million and solve for x.
[tex]10,000,000=10(2)^x[/tex][tex]\begin{gathered} \text{Divide both sides by 10.} \\ 1,000,000=2^x \\ Convert\text{ to logarithmic form.} \\ \log _21,000,000=x \\ x\approx19.93 \\ x\approx20 \end{gathered}[/tex]Thus, it will take approximately 20 years for the population of the mice to at least reach 10 million.
simplify (2^x-6)(^x-2)
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solution
[tex]\begin{gathered} (2\sqrt[]{x}-6)(\sqrt[]{x}-2) \\ 2\sqrt{x}\sqrt{x}+2\sqrt{x}\mleft(-2\mright)+\mleft(-6\mright)\sqrt{x}+\mleft(-6\mright)\mleft(-2\mright) \\ 2\sqrt{x}\sqrt{x}-2\cdot\: 2\sqrt{x}-6\sqrt{x}+6\cdot\: 2 \\ 2x-10\sqrt{x}+12 \end{gathered}[/tex]answer is:
[tex]2x-10\sqrt[]{x}+12[/tex]How do you find the common difference of an arithmetic sequence? Ex. 28,18,8,-2
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Answer:
To find the common difference of an arithmetic sequence, we compute the difference of an element of the sequence and its predecessor.
Example: Given the sequence 28, 18, 8, -2, the common difference is:
[tex]\begin{gathered} 18-28=-10, \\ 8-18=-10, \\ -2-8=-10. \end{gathered}[/tex]In a aquarium, 2/5 of the fish are surgeonfish. Of those surgeonfish, 3/4 are yellow tangs. What fraction of all the fish in the aquarium are not surgeonfish?
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2/5 of the fish are surgeonfish
3/4 of the surgeonfish are yellow tags-
To find the number of fishes that are not surgeonfish, subtract the number of surgeonfish to the total (1)
1 - 2/5 = 5/5 -2/5 = 3/5
3/5 of all the fish are not surgeonfish
A pawn is in the shape of a rectangle of a trapezoid with a height of 60 feet and based of 80 feet and 120 feet how many bags of fertilizer must be purchased to cover the lawn if each bag covers 4000 square feet ? Bags of fertilizer must be purchased to cover the lawn
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2 bags
Explanations:First, we need to get the area of the trapezoid. This is expressed as:
[tex]A=0.5(a+b)h[/tex]a and b are the bases
h is the height of the trapezoid
Given the following parameters
a = 80 feet
b = 120 feet
h = 60 feet
Substitute the given parameters into the formula:
[tex]\begin{gathered} A=0.5(120+80)\cdot60 \\ A=0.5\times200\times60 \\ A=6000ft^2 \end{gathered}[/tex]If each bag covers 4000 square feet of the lawn, the number of bags needed to be purchased is expressed as:
[tex]\begin{gathered} n=\frac{A}{4000} \\ n=\frac{6000}{4000} \\ n=1.5\text{bags} \\ n\approx2\text{bags} \end{gathered}[/tex]This shows that 2 bags of fertilizer must be purchased to cover the lawn (to the nearest bag)
In the figure shown, AD - BC, DALAC, and CBLBD.Prove ACADDBCExplain your reasoning,АBCBKA
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For this problem we will first prove that AC is congruent to BD then by the SSS criteria the triangles will are congruent.
To prove that AC is congruent to BD we use the pythagorean theorem for the right triangles CAD and DBC ( the theorem states that the sum of the square of the legs of the triangles is equal to the square of the hypotenuse):
[tex]\begin{gathered} BC^2+BD^2=DC^2 \\ \text{AD}^2+AC^2=DC^2 \\ \Rightarrow \\ BC^2+BD^2=AD^2+AC^2 \end{gathered}[/tex]Now, from the hypothesis we know that AD=BC ( this side is shared by both triangles) then
[tex]\begin{gathered} AD^2=BC^2 \\ \text{Substituting in the previous equation we get} \\ AD^2+BD^2=AD^2+AC^2 \\ \text{Cancelling AD}^2\text{ on each side we get } \\ BD^2=AC^2 \\ BD=AC \end{gathered}[/tex]Then from the SSS(side side side) criteria we have that triangle CAD is congruent to triangle DBC.
Compare the quantities in Column A and Column B Column A Column B The solutions of
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Given data:
The given inequality in the column A is 4x-30≥ -3x+12.
The given inequality in the column B is 1/2 x +3 < -2x-6.
The first inequality can be written as,
[tex]\begin{gathered} 4x-30\text{ }\ge\text{ -3x+12} \\ 7x\ge42 \\ x\ge6 \end{gathered}[/tex]The second inequality can be written as,
[tex]\begin{gathered} \frac{1}{2}x+3<-2x-6 \\ \frac{1}{2}x+2x<-9 \\ \frac{5}{2}x<-9 \\ x<-3.6 \end{gathered}[/tex]Thus, the quantity in the column A is always greater, so first option is correct.