The expression which is equivalent to the given expression; 2(3a+2b−7) is; 6a +4b -14
Equivalent expressionsThe given expression is;
2(3a+2b−7)Upon expansion by multiplying all elements in the parenthesis by 2; we have;
2(3a+2b−7) = 6a +4b -14Read more on equivalent expressions;
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Final answer:
By distributing the 2 across each term in the expression 2(3a+2b-7), the equivalent expression is 6a + 4b - 14, which corresponds to answer choice D.
Explanation:
To determine which expression is equivalent to 2(3a+2b-7), we need to distribute the 2 across each term in the parentheses. This means we'll multiply 2 by 3a, 2b, and -7 separately. Here's the step-by-step distribution:
2 × 3a = 6a2 × 2b = 4b2 × -7 = -14Combining these results, we get the equivalent expression: 6a + 4b - 14, which matches answer choice D.
If two lines are parallel, which statement must be true?
) A. The two lines have zero slopes.
B. The two lines have undefined slopes.
C. The two lines have the same slope.
D. The two lines have opposite slopes.
Answer:
c
Step-by-step explanation:
0 As shown in the diagram below, M, R, and T are
midpoints of the sides of ABC.
If AB = 18, AC = 14, and BC = 10, what is the
perimeter of quadrilateral ACRM?
1) 35
2) 32
24
4) 21
Answer:
The answer to your question is A.35
The perimeter of quadrilateral ACRM is 30. The closest option is 32 (Option 2).
Since M, R, and T are midpoints of the sides of ABC, they divide each side into two equal parts. Therefore, AM = MB, BR = RC, and CT = TA.
Now, let's find the lengths of AM, BR, and CT.
1. **Length of AM:**
[tex]\[ AM = \frac{1}{2} \cdot AB = \frac{1}{2} \cdot 18 = 9 \][/tex]
2. **Length of BR:**
[tex]\[ BR = \frac{1}{2} \cdot BC = \frac{1}{2} \cdot 10 = 5 \][/tex]
3. **Length of CT:**
[tex]\[ CT = \frac{1}{2} \cdot AC = \frac{1}{2} \cdot 14 = 7 \][/tex]
Now, we need to find the perimeter of quadrilateral ACRM:
[tex]\[ \text{Perimeter} = AM + BR + RC + CT \][/tex]
[tex]\[ \text{Perimeter} = 9 + 5 + 7 + 9 = 30 \][/tex]
So, the perimeter of quadrilateral ACRM is 30. The closest option is 32 (Option 2). Please double-check the answer choices, as the calculated perimeter is not exactly matching any of the provided options.
Do you use the greatest common factor or the least common multiple to add two fractions? In this example what is it? 4/5+5/6 least common multiple of 3 or greatest common factor 30 or least common multiple 30
Answer:
I'm pretty sure that you would try to find the least common multiple.
Step-by-step explanation:
Final answer:
When adding fractions, you need to find a common denominator using the least common multiple (LCM) of the denominators. In this case, the LCM of 5 and 6 is 30, so you multiply each fraction by a factor to make the denominators the same. Then, you can add the numerators together.
Explanation:
When adding fractions, you need to use the least common multiple (LCM) of the denominators to find a common denominator. In this example, the denominators are 5 and 6. The LCM of 5 and 6 is 30, so you need to find equivalent fractions with a denominator of 30.
To do this, multiply the numerator and denominator of each fraction by a factor that will result in a denominator of 30. For the first fraction, multiply both the numerator and denominator by 6 to get 24/30. For the second fraction, multiply both the numerator and denominator by 5 to get 25/30.
Finally, you can add the numerators together (24 + 25 = 49) and keep the common denominator of 30, giving you the final fraction of 49/30.
3. In a single growing season at the Smith Family Orchard, the average yield per apple tree is 150 apples when the number of trees per acre is 100. For each additional tree over 100, the average yield per tree decreases by 1.
a. What would be the average yield per tree if the number of trees per acre was doubled? What would be the total yield in that case?
b. How many trees should be planted per acre to maximize the total yield?
Answer:
A.10000
B.25 more trees must be planted
Step-by-step explanation:
⇒Given:
The intial average yield per acre [tex]y_{i}[/tex] = 150The initial number of trees per acre [tex]t_{i}[/tex] = 100For each additional tree over 100, the average yield per tree decreases by 1 i.e , if the number trees become 101 , the avg yield becomes 149.Total yield = (number of trees per acre)[tex]*[/tex](average yield per acre)A.
⇒If the total trees per acre is doubled , which means :
total number of trees per acre [tex]t_{f}[/tex] = [tex]2*t_{i}[/tex] = 200
the yield will decrease by : [tex]t_{f}[/tex] - [tex]y_{i}[/tex]
[tex]y_{f}= 150-100= 50[/tex]
⇒total yield = [tex]50*200=10000[/tex]
B.
⇒to maximize the yield ,
let's take the number of trees per acre to be 100+y ;
and thus the average yield per acre = 150 - y;
total yield = [tex](100+y)*(150-y)\\=15000+50y-y^{2} \\[/tex]
this is a quadratic equation. this can be rewritten as ,
⇒ [tex]=15000+50y-y^{2}\\=15000+625 - (625 - 50y +y^{2})\\=15625 - (y-25)^{2}[/tex]
In this equation , the total yield becomes maximum when y=25;
⇒Thus the total number of trees per acre = 100+25 =125;
The angle of elevation of a ladder is 19∘ and the ladder is 12 feet from the wall. What is the height of the ladder in feet?
Answer:
(24x)(19x2) 12/2 19
Step-by-step explanation:
(24x)(19x2) 12/2 19
Answer:
12.7 ft
Step-by-step explanation:
You're going to need to use acos
so
12 = acos (19) = 12.69
Then round up to 12.7
Please help me answer this.
Answer:
See explanation
Step-by-step explanation:
Statements Reasons
1. [tex]\overline{AD}\parallel \overline{BC}[/tex] Given
2. [tex]\angle ADB\cong \angle CBD[/tex] As alternate interior angles when parallel lines AD and BC intersect by ltransversal BD
3. [tex]\overline{AD}\cong \overline{BC}[/tex] Given
4. [tex]\overline{BD}\cong \overline{DB}[/tex] Reflexive property
5. [tex]\triangle ADB\cong \triangle CBD[/tex] SAS postulate
6. [tex]\angle ABD\cong \angle CDB[/tex] Corresponding parts of congruent triangles are congruent
7. [tex]\overline{AB}\parallel \overline{CD}[/tex] Inverse alternate interior angles theorem
If y=3x^2 - 2x +5
What is the average slope for this function between the points at:
i. x= -3 and x= -1
ii. x=-3 and x=0
iii. x= 1-h and x= 1+h
USE DIFFERENCE QUOTIENT FORMULA.
Answer:
Part i) -14
Part ii) 11
Part iii) 4
Step-by-step explanation:
we know that
The average rate of change or slope using the difference quotient formula is equal to
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
Part i) x= -3 and x= -1
In this problem we have
[tex]a=--3[/tex]
[tex]b=-1[/tex]
[tex]f(a)=f(-3)=3(-3)^{2} -2(-3)+5=38[/tex]
[tex]f(b)=f(-1)=3(-1)^{2} -2(-1)+5=10[/tex]
Substitute
[tex]\frac{10-38}{-1+3}[/tex]
[tex]\frac{-28}{2}[/tex]
[tex]-14[/tex]
Part ii) x= -3 and x= 0
In this problem we have
[tex]a=--3[/tex]
[tex]b=0[/tex]
[tex]f(a)=f(-3)=3(-3)^{2} -2(-3)+5=38[/tex]
[tex]f(b)=f(0)=3(0)^{2} -2(0)+5=5[/tex]
Substitute
[tex]\frac{5-38}{0+3}[/tex]
[tex]\frac{-33}{3}[/tex]
[tex]-11[/tex]
Part iii) x= (1-h) and x=(1+h)
In this problem we have
[tex]a=-(1-h)[/tex]
[tex]b=(1+h)[/tex]
[tex]f(a)=f(1-h)=3(1-h)^{2} -2(1-h)+5=3(1-2h+h^2)-2+2h+5=3-6h+3h^2+2h+3=3h^2-4h+6[/tex]
[tex]f(b)=f(1+h)=3(1+h)^{2} -2(1+h)+5=3(1+2h+h^2)-2-2h+5=3+6h+3h^2-2h+3=3h^2+4h+6[/tex]
Substitute
[tex]\frac{(3h^2+4h+6)-(3h^2-4h+6)}{1+h-(1-h)}[/tex]
[tex]\frac{8h}{2h}[/tex]
[tex]4[/tex]
1 in 4 adults are on a diet. in a random sample of 10 adults, what is the probability that the number on a diet is exactly 4
Answer:
14.6%
Step-by-step explanation:
Use binomial probability:
P = nCr pʳ (1−p)ⁿ⁻ʳ
where n is the number of trials,
r is the number of successes,
and p is the probability of success.
Here, n = 10, r = 4, and p = 1/4.
P = ₁₀C₄ (1/4)⁴ (1−1/4)¹⁰⁻⁴
P = 210 (1/4)⁴ (3/4)⁶
P ≈ 0.146
There is an approximate 14.6% probability that exactly 4 of the 10 adults are on a diet.
Simplify (-36x)(-1/3)(2/9)(-4y)
Answer:
[tex]\frac{32xy}{3}[/tex]
Step-by-step explanation:
What is -y+9z-16y-25z+4 simplified
The expression -y + 9z - 16y - 25z + 4 simplifies to -17y - 16z + 4 by combining like terms, which are the y terms and the z terms separately.
Explanation:The task is to simplify the expression -y + 9z - 16y - 25z + 4. To do this, we will combine like terms.
First, combine the terms that contain y:
-y - 16y = -17y
Then, combine the terms that contain z:
9z - 25z = -16z
The number without a variable, which is 4 in this case, remains the same since it does not have any like terms to combine with.
Now, we put the simplified terms together:
-17y - 16z + 4
The expression is now simplified to its least terms.
A survey on healthy choices was given to gym members at Work It Out. The results
showed that 90% of its members drank protein shakes. Of the members who drank
protein shakes, 30% took weight-loss medication whereas only 10% of members
who did not drink protein shakes took weight-loss medication. What is the
probability that a gym member does not take weight-loss medication?
1) 0.28
2) 0.66
3) 0.63
4) 0.72
Answer:
The correct option is 4) 0.72.
Step-by-step explanation:
Consider the provided information.
Let us consider that there are 100 members.
(Note: you can take any number the answer will remain the same.)
90% of its members drank protein shakes and of the members who drank protein shakes.
90% of 100 is 90.
Thus, 90 members drank protein shake.
Of the members who drank protein shakes, 30% took weight-loss medication.
30% of 90 is 27.
That means 27 out of 90 took weight loss medication.
Out of 100 members 90 drank protein shakes that means 10 members does not take protein shakes.
Only 10% of members who did not drink protein shakes took weight-loss medication.
10% of 10 is 1, that means 1 member out of 10 took weight loss medication.
Thus the required table is:
Medication Not Medication Total
Drank shakes 27 63 90
Doesn't Drank shakes 1 9 10
Total 28 72 100
Now we need to find the probability that a gym member does not take weight-loss medication.
72 out of 100 members does not take weight-loss medication.
Therefore, the required probability is: [tex]\frac{72}{100}=0.72[/tex]
Hence, the correct option is 4) 0.72.
2. The registration at a preschool is $125. Then,
parents must also pay $475 per month for
tuition. Write an equation to represent the
total cost after each month. Identify your
variables. a) What is the rate of change?
b) What is the initial value?
c) What is the Independent variable?
d) What is the dependent variable?
The equation of the total cost is y = 475 x + 125, where y is the total cost for x months
a) The rate of change is $475 per month
b) The initial value is $125
c) The independent variable is the number of months (x)
d) The dependent variable is the total cost after each month (y)
Step-by-step explanation:
The given is:
The registration at a preschool is $125The parents must also pay $475 per month for tuitionAssume that the total cost is y for x months
∵ The registration fee = $125 ⇒ paid once
∵ The parents pay $475 per month for tuition
∵ The number of months is x
∴ The total cost y = 475 x + 125
The equation of the total cost is y = 475 x + 125, where y is the total cost for x months
∵ y = 475 x + 125 is in the form of the linear equation y = m x + b,
m is the slope of the line and b is the y-intercept
∵ The slope of the line m is the rate of change y with respect to x
∴ m = 475
∴ The rate of change is $475 per month
a) The rate of change is $475 per month
∵ b is the initial value of y at x = 0
∴ b = 125
∴ The initial value is $125
b) The initial value is $125
∵ The independent variable is x
∴ The independent variable is the number of months (x)
c) The independent variable is the number of months (x)
∵ The dependent variable is y
∴ The dependent variable is the total cost after each month (y)
d) The dependent variable is the total cost after each month (y)
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you jus drove your car 450 miles and used 50 gallons of gas. you know that the gas tank on your car holds 15 1/2 gallons of gas. what is the most number of miles you can drive on one tank of gas?
Final answer:
To find the most number of miles you can drive on one tank of gas, divide the total miles driven by the total gallons used to calculate the miles per gallon (MPG). Then, multiply the MPG by the number of gallons in the gas tank to find the maximum number of miles that can be driven on one tank.
Explanation:
To find the most number of miles you can drive on one tank of gas, you can use the given information. You know that you just drove your car 450 miles and used 50 gallons of gas. Therefore, to find the miles per gallon (MPG), divide the total miles driven by the total gallons used. In this case, 450 miles divided by 50 gallons gives you an MPG of 9. This means that for every gallon of gas, your car can travel 9 miles.
Now, you know that the gas tank on your car holds 15 1/2 (or 15.5) gallons of gas. To find the maximum number of miles you can drive on one tank of gas, multiply the MPG by the number of gallons in the tank. So, 9 MPG multiplied by 15.5 gallons equals 139.5 miles. Therefore, the most number of miles you can drive on one tank of gas is 139.5 miles.
The most number of miles you can drive on one tank of gas is 139.5 miles.
To determine the most number of miles you can drive on one tank of gas, follow these steps:
Calculate the car's average miles per gallon (mpg):
You drove 450 miles using 50 gallons of gas.
The average mpg is calculated by dividing the total miles driven by the total gallons of gas used: [tex]\frac{450 \text{ miles}}{50 \text{ gallons}} = 9 \text{ mpg}.[/tex]
Determine the capacity of one full tank of gas in your car:
The gas tank holds 15 1/2 gallons of gas. Convert 15 1/2 to a decimal: [tex]15 \frac{1}{2} = 15.5 \text{ gallons}.[/tex]
Calculate the most number of miles you can drive on one full tank of gas:
Multiply the car's average mpg by the gas tank's capacity: [tex]9 \text{ mpg} \times 15.5 \text{ gallons} = 139.5 \text{ miles}.[/tex]
8 - (4/5)x - 14 - 2x
For this case we must simplify the following expression:
[tex]8- \frac {4} {5} x-14-2x =[/tex]
We add similar terms:
[tex]8-14- \frac {4} {5} x-2x =[/tex]
We take into account that:
Equal signs are added and the same sign is placed.
Different signs are subtracted and the major sign is placed.
[tex]-6+ (\frac {-4-10} {5} x) =\\-6+ (\frac {-14} {5} x) =\\-6- \frac {14} {5} x[/tex]
Finally, the simplified expression is:
[tex]-6- \frac {14} {5} x[/tex]
ANswer:
[tex]-6- \frac {14} {5} x[/tex]
What is 30.7 rounded to the nearest ones place
Answer:
its 31 when you round to 2 decimal points.
. The circumference of a circle is 55/7
. What is the diameter of the circle.
Answer: 55/7π
Step-by-step explanation:
circumference = 2πr
2πr = 55/7
r = 55/14π
d = 2r
d = 55/7π
Translate the description as an algebraic expression:
the square of the ratio of 11 and k
Answer:
The given description is equivalent to the algebraic expression,
[tex](\frac {11}{k})^{2}[/tex]
Step-by-step explanation:
The given description is ,
'The square of the ratio of 11 and k'
which is equivalent to the algebraic expression,
[tex](\frac {11}{k})^{2}[/tex]
The table shows the distance a runner covers during a race:
Time, x
(minutes)
Distance, y
(miles)
5 0.5
10 1
15 1.5
20 2
25 2.5
Which equation represents the relationship between x and y?
a. y = 0.5x
b. y = 10x
c. y = x + 10
d. y = 110x
win brainliest
Answer:
b. y=10x
Step-by-step explanation:
here, x represents time(minutes) : 0.51
1.5
2
2.5
y represents distance (miles): 5
10
15
20
25
to write any equation(linear) basically we need 2 points lying on itlet the 2 points be: (1,10) and (2,20) [from the given information]
now, let (A,B) and (C,D) are two points on a linethen the equation of it can be given as :
[tex]y-B=(\frac{D-B}{C-A} )*(x-A)[/tex]
so, here the required equation will be,[tex]y-10=(\frac{20-10}{2-1} )*(x-1)\\[/tex]
this is also equal to, [tex]y-10=10*(x-1)\\y-10=10x-10\\y =10x-10+10\\y=10x[/tex]
therefore, option c represents the relationship between x and y.[trick: when you have options, you can directly substitute x and respective y values and check if the equation satisfies it. the equation which satisfies all the x and respective y values will be the correct equation.]how many liters each of a 60% acid solution and a 80% acid solution must be used to produce 80 liters of a 75% acid solution
Answer:
20 liters of 60% acid solution and 60 liters of 80% acid solution
Step-by-step explanation:
Let the amount of 60% solution needed be "x", and
amount of 80% solution needed be "y"
Since we are making 80 liters of total solution, we can say:
x + y = 80
Now, from the original problem, we can write:
60% of x + 80% of y = 75% of 80
Converting percentages to decimals by dividing by 100 and writing the equation algebraically, we have:
0.6x + 0.8y = 0.75(80)
0.6x + 0.8y = 60
We can write 1st equation as:
x = 80 - y
Now we substitute this into 2nd equation and solve for y:
0.6x + 0.8y = 60
0.6(80 - y) + 0.8y = 60
48 - 0.6y + 0.8y = 60
0.2y = 12
y = 12/0.2
y = 60
Also, x is:
x = 80 - y
x = 80 - 60
x = 20
Thus, we need
20 liters of 60% acid solution and 60 liters of 80% acid solution
Jadon rode in the car 42 miles. This represents 70% of his entire trip. What is the total number of miles in jadons trip?
There are 60 miles in Jadon's trip.
Step-by-step explanation:
Distance covered by Jadon = 42 miles
This is 70% of his entire trip.
Let,
x be the total number of miles in Jadon's trip.
We know that,
70% of x = 42 miles
[tex]\frac{70}{100}*x=42\\0.7x=42[/tex]
Dividing both sides by 0.07;
[tex]\frac{0.7x}{0.7}=\frac{42}{0.7}\\x=60\ miles[/tex]
There are 60 miles in Jadon's trip.
Keywords: distance, percentage
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Mrs. Aviles is planning a fruit-cup party for a class of 18 students and two teachers. The delivery charge is $10 and fruit cups cost $3 each. What is the range of the function?
Answer:
The cost of the 20 fruit cup is $ 60 .
Step-by-step explanation:
Given as :
The total students for fruit cup party = 18
The total teachers for fruit cup party = 2
The cost of each fruit cup =$ 3
The delivery charge = $ 10
So, The total number of people for party = Total number of student + Total number of teachers
Or , The total number of people for party = 18 + 2 = 20
∵ each fruit cup cost $ 3
∴ The cost of 20 fruit cup = $ 3 × 20
I.e The cost of 20 fruit cup = $ 60
Hence The cost of the 20 fruit cup is $ 60 . Answer
The range of the function for the fruit-cup party is $10 to $70, including the delivery charge and the cost of the fruit cups.
Explanation:To find the range of the function for the fruit-cup party, we need to consider the total cost of the fruit cups and the delivery charge. The delivery charge is a fixed cost of $10. The cost of each fruit cup is $3. So, the total cost of the fruit cups for the 18 students and 2 teachers is (18 + 2) × $3 = $60. Therefore, the range of the function is $10 to $70, which includes the delivery charge and the total cost of the fruit cups.
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what does 2+2 equal to
Answer:
Step-by-step explanation:
1
2+2=4 because you add it ok well bye :)
Ajay sells two cycles at 2500$ each. If he makes a profit of 25% on one and incurs a loss of 25% on the other, find his net profit or loss percent.
Answer:
The total loss percentage is 6.25%.
Step-by-step explanation:
Let the cost price of the first cycle is $x.
So, for the first cycle, 25% profit is there for selling the cycle at $2500.
Then, [tex]x(1 + \frac{25}{100}) = 2500[/tex]
⇒ 1.25x = 2500
⇒ x = $2000
Again, let us assume that the cost price of the second cycle is $y.
So, by selling the cycle at $2500 there is a loss of 25%.
Therefore, [tex]y(1 - \frac{25}{100} ) = 0.75y = 2500[/tex]
⇒ y = $3333.33
Therefore, the total cost price of two cycles = $(2000 + 3333.33) = $5333.33 and total selling price is $(2500 + 2500) = $5000.
Therefore, the total loss percentage is [tex]\frac{5333.33 - 5000}{5333.33} \times 100 = 6.25[/tex] % (Answer)
Lynn has 4 stacks of quarters.There are 8 quarters in each stack. How many quarters does Lynn have?
Answer:
32
Step-by-step explanation:
4 x 8 = 32
Answer:
32 quarters
Step-by-step explanation:
if there are 4 stack with 8 quarters in each stack, then we know that all we have to do is multiply 4 x 8 which equals 32
can the side lengths 12,15, and 13 form a triangle?
Answer:
Yes
Step-by-step explanation:
It is a Pythagorean Triple. A Pythagorean Triple is the sides of a triangle that fit perfectly into the Pythagorean Theorem, which is:
a²+b²=c²
12, 13, and 15 are numbers that you should know off the top of your head so if you see a triangle with 2 of those numbers, you instantly know the 3rd number.
~Stay golden~ :)
Answer: Yes
Step-by-step explanation: To determine if these side lengths can form a triangle, I attached a rule in the image provided which is very helpful to look at especially when you're new at this.
If a triangle has sides with lengths of 12, 15, and 13, notice that 12 + 15 or 27 is greater than 13.
So the sum of the lengths of two sides of the triangle is greater than the length of the third side.
This means that the triangle with sides of lengths of 12, 15, and 13, is possible.
If tan∅ = √15÷10, find cot∅
Answer:
cot(∅) = [tex]\frac{10\sqrt{15} }{15}[/tex]
Step-by-step explanation:
tan ∅ and cot ∅ are inverse functions
therefore the inverse of
tan ∅ = √15 / 10
is equal to
cot ∅ = 10 / √15
Rationalizing the denominator
[tex]\frac{10}{\sqrt{15} }[/tex] * [tex]\frac{\sqrt{15} }{\sqrt{15} }[/tex]
cot ∅ = [tex]\frac{10\sqrt{15} }{15}[/tex]
Answer:(2\sqrt(15))/(3)
Step-by-step explanation:
what is the solution to this inequality 7+x>5
Answer:
x > - 2
Step-by-step explanation:
Given
7 + x > 5 ( subtract 7 from both sides )
x > - 2
8 copies of the same CD cost $120.What is the cost of one CD?
Answer:
$15
Step-by-step explanation:
120/8=15
Answer:
Step-by-step explanation:
Total Cost of 8 copies of same CD =$120
Because the copies of the CD is same, the cost of each one should be same
Cost of one CD should be $120/8=$15
A 2-column table has 5 rows. The first column is labeled x with entries negative 2, negative 1, 0, 1, 2. The second column is labeled f (x) with entries 12.5, 2.5, 0.5, 0.1, 0.02. Which exponential function is represented by the table?
Answer:
[tex]f(x)=0.5(0.2)^x[/tex]
Step-by-step explanation:
Exponential function can be in this form : [tex]f(x)=ab^x[/tex]
To find a and b we can use any two pairs from the table.
We can use (0, 0.5) and (1,0.1)
[tex]0.5=a*b^0\\a=0.5[/tex]
let's find b
[tex]0.1=0.5b^1\\b=0.2[/tex]
The exponential function represented by the table is [tex]\( f(x) = 3.125 \times (0.5)^x \)[/tex], where x can take values -2, -1, 0, 1, or 2, and f(x) corresponds to the values given in the table.
1. Examine the given table with two columns: x and f(x), where x takes values -2, -1, 0, 1, and 2, and f(x) takes corresponding values 12.5, 2.5, 0.5, 0.1, and 0.02.
2. Notice that as x increases by 1, f(x) decreases by a factor of 5 (12.5 / 2.5 = 5, 2.5 / 0.5 = 5, etc.), indicating an exponential decay pattern.
3. Write the general form of an exponential decay function:
[tex]\[ f(x) = a \times b^x \][/tex]
where a is the initial value and b is the decay factor.
4. Use the first row of the table (x = -2, f(x) = 12.5) to find the initial value, a:
[tex]\[ 12.5 = a \times (0.5)^{-2} \] \[ 12.5 = a \times 4 \] \[ a = \frac{12.5}{4} = 3.125 \][/tex]
5. Substitute the initial value, a, into the exponential function:
[tex]\[ f(x) = 3.125 \times (0.5)^x \][/tex]
Therefore, the exponential function represented by the table is [tex]\( f(x) = 3.125 \times (0.5)^x \)[/tex], where x can take values -2, -1, 0, 1, or 2, and f(x) corresponds to the values given in the table.
Calibrating a scale:
Making sure that the scales used by businesses in the United States are accurate is the responsibility of the National Institute for Standards and Technology (NIST) in Washington, D.C. Suppose that NIST technicians are testing a scale by using a weight known to weigh exactly 1000 grams. The standard deviation for scale reading is known to be o 2.8. They weigh this weight on the scale 50 times and read the result each time. The 50 scale readings have a sample mean of x = 1001.1 grams. The calibration point is set too high if the mean scale reading is more than 1000 grams. The technicians want to perform a hypothesis test to determine whether the calibration point is set too high.
Use a = 0.01 level of significance and the critical value method.
(a) State the appropriate null and alternate hypotheses.
H0:
H1:
This hypothesis test is a left, right, or two tailed?
Answer:
We conclude that the calibration point is set too high.
Step-by-step explanation:
We are given the following in the question:
Population mean, μ = 1000 grams
Sample mean, [tex]\bar{x}[/tex] = 1001.1 grams
Sample size, n = 50
Alpha, α = 0.05
Population standard deviation, σ = 2.8 grams
First, we design the null and the alternate hypothesis
[tex]H_{0}: \mu = 1000\text{ grams}\\H_A: \mu > 1000\text{ grams}[/tex]
We use One-tailed(right) z test to perform this hypothesis.
Formula:
[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]
Putting all the values, we have
[tex]z_{stat} = \displaystyle\frac{ 1001.1 - 1000}{\frac{2.8}{\sqrt{50}} } = 2.778[/tex]
Now, [tex]z_{critical} \text{ at 0.01 level of significance } = 2.326[/tex]
Since,
[tex]z_{stat} > z_{critical}[/tex]
We reject the null hypothesis and accept the alternate hypothesis. We accept the alternate hypothesis. We conclude that the calibration point is set too high.
Final answer:
The null hypothesis is that the calibration point is set correctly and the alternate hypothesis is that the calibration point is set too high.
Explanation:
The appropriate null and alternate hypotheses for this hypothesis test are:
H0: The calibration point is set correctly (mean scale reading = 1000 grams)
H1: The calibration point is set too high (mean scale reading > 1000 grams)
This hypothesis test is a right-tailed test because the alternate hypothesis is testing for a value greater than the null hypothesis.