The equation for Harry's hunger in terms of time can be written as,H(t) = 7.5.cos(π.t) + 22.5
Given:Hunger of Harry as a function of time,H(t)H(t) can be modeled by a sinusoidal expression of the form,a⋅cos(b⋅t)+da⋅cos(b⋅t)+d, where Harry wakes up at t=0t=0t=0, his hunger is at a maximum, and he desires 30 kg 30 kg30, start text, space, k, g, end text of pigs.
Within 2 22 hours, his hunger subsides to its minimum, when he only desires 15 kg 15 kg15, start text, space, k, g, end text of pigs.
Therefore, the equation of the form for H(t)H(t) will be,H(t) = A.cos(B.t) + C where, A is the amplitude B is the frequency (number of cycles per unit time)C is the vertical shift (or phase shift)
Thus, the maximum and minimum hunger of Harry can be represented as,When t=0t=0t=0, Harry's hunger is at maximum, i.e., H(0)=30kgH(0)=30kg30, start text, space, k, g, end text.
When t=2t=2t=2, Harry's hunger is at the minimum, i.e., H(2)=15kgH(2)=15kg15, start text, space, k, g, end text.
According to the given formula,
H(t) = a.cos(b.t) + d ------(1)Where a is the amplitude, b is the angular frequency, d is the vertical shift.To find the value of a, subtract the minimum value from the maximum value.a = (Hmax - Hmin)/2= (30 - 15)/2= 15/2 = 7.5To find the value of b, we will use the formula,b = 2π/period = 2π/(time for one cycle)The time for one cycle is (2 - 0) = 2 hours.
As Harry's hunger cycle is a sinusoidal wave, it is periodic over a cycle of 2 hours.
Therefore, the angular frequency,b = 2π/2= π
Therefore, the equation for Harry's hunger in terms of time can be written as,H(t) = 7.5.cos(π.t) + 22.5
Answer: H(t) = 7.5.cos(π.t) + 22.5.
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⦁ The construction of copying is started below. The next step is to set the width of the compass to the length of . How does this step ensure that the new angle will be congruent to the original angle?
Answer:
i believe by creating radii of equal lengths.
Step-by-step explanation:
it gives a path to create an angle congruent to angle APB. The angle APB would have the same radii (BP and AP) and the same width as the congruent angle that would be created.
Wish you good luck.
21. In each of these problems, determine a suitable form for Y (t) if the method of undetermined coefficients is to be used. Do not evaluate the constants.
a. y"" - 2y" + y' = t³ + 2e^t
b. y''' - y' = te^-t + 2cost
c. y^4 - 2y'' + y = e^t + sin(t)
d. y^4 + 4y" = sin 2t + te^t + 4
e. y^4 - y''' - y" + y' = t² + 4 + tsin(t)
f. y^4 + 2y''' + 2y" = 3e^t + 2te^-t + e^-t sin(t)
Answer:
a. Y(t) = At³ + Be^t + Ct² + Dt + E
b. Y(t) = At + B + Ce^t + Dsin(t) + Ecos(t)
c. Y(t) = Aet + Bte^t + Csin(t) + Dcos(t)
d. Y(t) = At³ + Bt² + Ct + D + Ecos(2t) + Fsin(2t)
e. Y(t) = At² + Bt + C + Dsin(t) + Ecos(t) + Fsin(t) + Gcos(t)
f. Y(t) = Aet + Bte^-t + Ccos(t) + Dsin(t) + E + Ft + G
1. The annual sale volumes of three products X, Y, Z whose sale prices per unit are GHS 3.50, GHS 2.75, GHS 1.50 respectively, in two different markets I and II are shown below: Product Market X Y Z I 6000 9000 1300 II 12000 6000 17000 Find the total revenue in each market with the help of matrices.
Answer:
Step-by-step explanation:
To find the total revenue in each market, we can calculate the product of the sale volumes and sale prices per unit using matrices.
Let's represent the sale volumes as a matrix V and the sale prices per unit as a matrix P:
V = [6000 9000 1300]
[12000 6000 17000]
P = [3.50]
[2.75]
[1.50]
To calculate the total revenue in each market, we need to perform matrix multiplication between V and P, considering the appropriate dimensions. The resulting matrix will give us the total revenue for each product in each market.
Total revenue = V * P
Calculating the matrix multiplication:
[6000 9000 1300] [3.50] = [Total revenue for product X in Market I Total revenue for product Y in Market I Total revenue for product Z in Market I]
[12000 6000 17000] [2.75] [Total revenue for product X in Market II Total revenue for product Y in Market II Total revenue for product Z in Market II]
Performing the calculation:
[60003.50 + 90002.75 + 13001.50] = [Total revenue for product X in Market I Total revenue for product Y in Market I Total revenue for product Z in Market I]
[120003.50 + 60002.75 + 170001.50] [Total revenue for product X in Market II Total revenue for product Y in Market II Total revenue for product Z in Market II]
Simplifying the calculation:
[21000 + 24750 + 1950] = [Total revenue for product X in Market I Total revenue for product Y in Market I Total revenue for product Z in Market I]
[42000 + 16500 + 25500] [Total revenue for product X in Market II Total revenue for product Y in Market II Total revenue for product Z in Market II]
[47650] = [Total revenue for product X in Market I Total revenue for product Y in Market I Total revenue for product Z in Market I]
[84000] [Total revenue for product X in Market II Total revenue for product Y in Market II Total revenue for product Z in Market II]
Therefore, the total revenue in Market I is GHS 47,650 and the total revenue in Market II is GHS 84,000.
An important part of the customer service responsibilities of a company relates to the speed with which troubles in residential service can be repaired. Suppose that past data indicate that the likelihood is 0.70 that service troubles can be repaired on the same day they are reported. For the first five troubles reported on a given day, what is the mean of the distribution used in this problem?
Select one:
A.
350
B.
3.5
C.
0.35
D.
0.7143
E.
3500
The mean of the distribution used in this problem is 3.5.The answer is B. 3.5.
To find the mean of the distribution used in this problem, we need to calculate the expected number of troubles that can be repaired on the same day out of the first five reported. The likelihood of a trouble being repaired on the same day is given as 0.70.
The expected number of troubles repaired on the same day can be calculated by multiplying the probability of each event by the number of trials. In this case, we have five trials (the first five troubles reported) and a probability of 0.70 for each trial.
Expected number of troubles repaired on the same day = Probability of repair on the same day * Number of trials
= 0.70 * 5
= 3.5
This means that, on average, out of the first five troubles reported, we can expect 3.5 of them to be repaired on the same day based on the past data and the given likelihood. So, the correct answer is B. 3.5.
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Determine if (4,1) is a solution for the system of equations. Explain answer
y = -x + 5
y = 2x - 7
Answer:
Step-by-step explanation:
To determine if the point (4,1) is a solution for the system of equations, we need to substitute the values of x and y from the point into both equations and check if the equations hold true.
For the first equation, y = -x + 5, we substitute x = 4 and y = 1:
1 = -(4) + 5
1 = -4 + 5
1 = 1
The equation holds true for the first equation.
For the second equation, y = 2x - 7, we substitute x = 4 and y = 1:
1 = 2(4) - 7
1 = 8 - 7
1 = 1
The equation also holds true for the second equation.
Since the point (4,1) satisfies both equations, it is indeed a solution to the system of equations.
Suppose a finite population has 6 items and 2 items are selected at random without replacement,then all possible samples will be:
Select one:
a. 15
b. 2
c. 36
d. 6
e. 12
Note: Answer D is NOT the correct answer. Please find the correct answer. Any answer without justification will be rejected automatically.
When 2 items are selected without replacement from a population of 6 items, there are 15 possible samples that can be formed. Option A.
To determine the number of possible samples when 2 items are selected at random without replacement from a population of 6 items, we can use the concept of combinations.
The number of combinations of selecting k items from a set of n items is given by the formula C(n, k) = n! / (k! * (n-k)!), where n! represents the factorial of n.
In this case, we have a population of 6 items and we want to select 2 items. Therefore, the number of possible samples can be calculated as:
C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5 * 4!) / (2! * 4!) = (6 * 5) / (2 * 1) = 15. Option A is correct.
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What is -2.93(b + 12) = -11.72
What is b
(Solve two-step linear equations)
13. Tonia and Trinny are twins. Their friends give them identical cakes for their birthday. Tonia eats ⅛ of her cake and Trinny eats ⅙ of her cake. How much cake is left?
Answer:
If Tonia eats 1/8 of the cake, then the fraction of the cake left is:
1 - 1/8 = 7/8
If Trinny eats 1/6 of the cake, then the fraction of the cake left is:
1 - 1/6 = 5/6
Since Tonia and Trinny have identical cakes, the amount of cake left is the same for both of them. Therefore, the amount of cake left is:
(7/8 + 5/6) / 2 = 41/48
So there is 41/48 of the cake left.
Subject offered monthly charges Duration
(Subjects) (monthly charges) (duration)
Mathematics R280
Consumer Studies R350
8 hours Per month
3 hours per Saturday.
write down (in Simplified form) the ratio of the
amount charged in mathematics to consumer
studies
The ratio of the amount charged in Mathematics to Consumer Studies is 4:5.
To find the ratio of the amount charged in Mathematics to Consumer Studies, we need to divide the amount charged in Mathematics by the amount charged in Consumer Studies.
The amount charged in Mathematics is R280 per month, while the amount charged in Consumer Studies is R350 per month.
Therefore, the ratio of the amount charged in Mathematics to Consumer Studies can be calculated as:
280 / 350
To simplify this ratio, we can divide both the numerator and denominator by their greatest common divisor, which in this case is 70.
Dividing 280 by 70 gives us 4, and dividing 350 by 70 gives us 5.
So, the simplified ratio of the amount charged in Mathematics to Consumer Studies is:
4/5.
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Seafloor rocks from a secret area above the Arctic Circle will recently analyzed by the globe. The company had a contract with spacegov.bids to test the rocks for nickel. Zieglow found that setting samples from the first location were composed of an average of 8.43% nickel. Six samples from the second location yielded an average of 7.81% nickel. What was the overall average nickel content of the rock samples
Answer:
The overall average nickel content of the rock samples is approximately 7.97%.
Step-by-step explanation:
To find the overall average nickel content of the rock samples, we need to take into account the number of samples from each location. Since we know the average nickel content of each set of samples, we can use a weighted average formula:
overall average nickel content = (total nickel content from first location + total nickel content from second location) / (total weight of samples from both locations)
To calculate the total nickel content from each location, we need to multiply the average nickel content by the number of samples:
total nickel content from first location = 8.43% x 1 sample = 8.43%
total nickel content from second location = 7.81% x 6 samples = 46.86%
To calculate the total weight of the samples from both locations, we need to add up the number of samples:
total weight of samples from both locations = 1 + 6 = 7
Now we can substitute these values into the formula and calculate the overall average nickel content:
overall average nickel content = (8.43% + 46.86%) / 7 ≈ 7.97%
Therefore, the overall average nickel content of the rock samples is approximately 7.97%.
Question 5 of 8
Which choice is the solution set of the inequality below?
OA. x< 4.1
OB. X<-4.1
OC. X> 4.1
O D. x≤ 4.1
OE. -4.1
OF. x<-4.1 or x> 4.1
x < 4.1, as it represents the solution set for the given inequality. A.
The solution set of the inequality x < 4.1 can be determined by examining the given answer choices:
OA. x < 4.1:
This choice represents all values of x that are strictly less than 4.1.
It is a valid solution set for the given inequality.
OB. x < -4.1:
This choice represents all values of x that are strictly less than -4.1.
It is not a valid solution set for the given inequality.
OC. x > 4.1:
This choice represents all values of x that are strictly greater than 4.1.
It is not a valid solution set for the given inequality.
OD. x ≤ 4.1:
This choice represents all values of x that are less than or equal to 4.1.
It is not a valid solution set for the given inequality because the inequality is strict (x < 4.1), not inclusive.
OE. -4.1 < x < 4.1:
This choice represents all values of x that are strictly between -4.1 and 4.1.
It is not a valid solution set for the given inequality because it does not include the possibility of x being less than -4.1 or greater than 4.1.
OF. x < -4.1 or x > 4.1:
This choice represents two separate solution sets: one for x < -4.1 and another for x > 4.1.
It is not a valid solution set for the given inequality because it combines two separate possibilities.
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20 Points N Brainly Promised
The coterminal of 4π/3 angle measure are 10π/3 and -2π/3
How do you find angles that are coterminal with an angle measure of 4π/3?We add or subtract integer multiples of 2π.
So, 4π/3 + 2π = 10π/3 is one coterminal angle, and
4π/3 - 2π = -2π/3 is another coterminal angle.
To convert 125.67° to degree, minute, and second measure:
125.67° = 125° + 0.67°
Since there are 60 minutes in 1 degree, we can multiply 0.67 by 60 to get the minutes:
0.67° × 60 = 40.2'
Since there are 60 seconds in 1 minute, we can multiply 0.2 by 60 to get the seconds:
0.2' × 60 = 12"
So, 125.67° is equivalent to 125° 40.2' 12".
To find the degree measure equivalent to 10π/13 rounded to the nearest hundredth of a degree, we can divide 10π/13 by π and then multiply by 180° to convert from radians to degrees:
(10π/13) / π * 180° ≈ 138.46° (rounded to the nearest hundredth).
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What else would need to be congruent to show that ABC=AXYZ by SAS?
A
B
OA. ZB=LY
B. BC = YZ
OC. C= LZ
OD. AC = XZ
с
X
Z
Given:
AB XY
BC=YZ
What is needed to be congruent to show that ABC=AXYZ is AC ≅ XZ. option D
How to determine the statementGiven that in ΔABC and ΔXYZ, ∠X ≅ ∠A and ∠Z ≅ ∠C.
We are to select the correct condition that we will need to show that the triangles ABC and XYZ are congruent to each other by ASA rule..
ASA Congruence Theorem: Two triangles are said to be congruent if two angles and the side lying between them of one triangle are congruent to the corresponding two angles and the side between them of the second triangle.
In ΔABC, side between ∠A and ∠C is AC,
in ΔXYZ, side between ∠X and ∠Z is XZ.
Therefore, for the triangles to be congruent by ASA rule, we must have AC ≅ XZ.
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JLK is similar to PQR find the value of X
Answer:
30
Step-by-step explanation:
22/33=20/x
cross multiply
22x=33x20
22x=660
x=660/22
x=30
Is x=-4, x=1 parallel lines?
Step-by-step explanation:
Use the following models to show the equivalence of the fractions 35 and 610 a) Set modelUse the following models to show the equivalence of the fractions 35 and 610 a) Set model
Answer:
Step-by-step explanation:
no
they would be on different sides on the y axis
What are the coordinates of the image of point (−1, 5) after a counterclockwise rotation of 90° about the origin?
Responses
(1, 5)
(5, 1)
(−5, −1)
(-5, -1)
Answer: (5, -1)
Step-by-step explanation:
To rotate a point counterclockwise by 90° about the origin, we swap the x and y coordinates and negate the new x-coordinate. For the point (-1, 5), we swap the x and y coordinates to get (5, -1). The x-coordinate becomes positive, and the y-coordinate becomes negative. Therefore, the coordinates of the image of the point (-1, 5) after a counterclockwise rotation of 90° about the origin are (5, -1).
I think you put down the same answer choice twice and instead meant to say (5, -1) instead of (-5, -1) twice.
A random number from 1 to 5 is selected 50 times. The number 1 is selected 13 times, 2 is selected 8 times, 3 is selected 14 times, 4 is selected 6 times, and 5 is selected 9 times. What is the relative frequency of selecting a 2? Express your answer as a percent.
Answer:
Relative frequency of selecting a 2 = 8/50 = 0.16 = 16%
Step-by-step explanation:
You are selecting a random number between 1 and 5, and you perform this task 50 times.
Out of these 50 times, the outcome "2" appears 8 times.
Therefore the relative frequency of selecting the number 2 is:
f(2) = 8/50 = 0.16 which is 16%
On a coordinate plane, a curved line, labeled D, with a minimum value of (3, negative 8), labeled C, with x-intercept (1, 0), labeled B, and y-intercept (0, 10), labeled A. Match the letter of the key feature on the graph with its name. x-intercept: y-intercept: Relative minimum: Increasing interval:
The interval for the graphed function that has a local minimum of 0 is [-2, 0].
To determine the interval for the graphed function that has a local minimum of 0, we need to examine the behavior of the graph around the x-values where it crosses the x-axis.
We know that the graph crosses the x-axis at -2.5, 0, and 3. Since the local minimum occurs at 0, we need to find the interval where the graph is decreasing to the left of 0 and increasing to the right of 0.
From the given information, we can determine the following:
The graph is decreasing between -2.5 and 0 because it crosses the x-axis at -2.5 and 0, and the minimum value is at (-1.56, -6).
The graph is increasing between 0 and 3 because it crosses the x-axis at 3, and the maximum value is at (1.2, 2.9).
The correct option is [–2, 0].
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Question
On a coordinate plane, a curved line with minimum values of (negative 1.56, negative 6) and (3, 0), and a maximum value of (1.2, 2.9), crosses the x-axis at (negative 2.5, 0), (0, 0), and (3, 0), and crosses the y-axis at (0, 0). Which interval for the graphed function has a local minimum of 0? [–3, –2] [–2, 0] [1, 2] [2, 4]
Circumference of circle inscribed or circumscribed polygon
Hint: you will need to find the diameter of the circle, use Pythagorean Theorem)
ind then I out of the 3 problems.
Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.
8 cm
15 cm
The exact circumferences of the inscribed and circumscribed circles for the given polygons are 8π cm and 15π cm, respectively.
To find the exact circumference of a circle inscribed or circumscribed by a polygon, we can use the Pythagorean theorem to determine the diameter of the circle.
In the case of an inscribed polygon, the diameter of the circle is equal to the diagonal of the polygon. Let's consider the polygon with a diagonal of 8 cm. If we draw a line connecting two non-adjacent vertices of the polygon, we get a diagonal that represents the diameter of the inscribed circle.
Using the Pythagorean theorem, we can find the length of this diagonal. Let's assume the sides of the polygon are a and b. Then the diagonal can be found using the equation: diagonal^2 = a^2 + b^2. Substituting the given values, we have 8^2 = a^2 + b^2. Solving this equation, we find that a^2 + b^2 = 64.
For the circumscribed polygon with a diagonal of 15 cm, the diameter of the circle is equal to the longest side of the polygon. Let's assume the longest side of the polygon is c. Therefore, the diameter of the circumscribed circle is 15 cm.
Once we have determined the diameter of the circle, we can calculate its circumference using the formula C = πd, where C is the circumference and d is the diameter.
For the inscribed circle, the circumference would be C = π(8) = 8π cm.
For the circumscribed circle, the circumference would be C = π(15) = 15π cm.
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Copy the axes below.
a) By completing the tables of values to help
you, plot the lines y = 2x + 1 and
y = 10x on your axes.
b) Use your diagram to find the solution to the
simultaneous equations y = 2x + 1 and
y = 10 - x.
y = 2x+1
x012
Y
y = 10-x
x012
Y
Y
-3 -2 -1
10
2987
65
6
-5
4
3
NW
21
1
-14
--2
73
1 2 3 4 5 6 7 8 9 10 x
The solution to the simultaneous equations is x = 3 and y = 7
Finding the solution to the simultaneous equationsFrom the question, we have the following parameters that can be used in our computation:
y = 2x + 1
y = 10 - x
Subtract the equations
So, we have
3x - 9 = 0
This gives
3x = 9
So, we have
x = 3
Next, we have
y = 10 - x
y = 10 - 3
Evaluate
y = 7
Hence, the solution is x = 3 and y = 7
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Team A and Team B together won 50% more games than Team C did. Team A won 50% as many games as Team B did. The three teams won 60 games in all. How many games did each team win?
Vinay buys some fruits. He buys 7 fruits more than the place value of 2 in the number 37,523. Find out the number of fruits that vinay buys and write the same in number names.
Vinay buys "two thousand seven" fruits.
To find the number of fruits that Vinay buys, we need to determine the place value of 2 in the number 37,523 and add 7 to it.
In the number 37,523, the digit 2 is in the thousands place.
The place value of 2 in the thousands place is 2,000.
Adding 7 to the place value of 2, we get:
2,000 + 7 = 2,007.
Therefore, Vinay buys 2,007 fruits.
In number names, we can write 2,007 as "two thousand seven."
So, Vinay buys "two thousand seven" fruits.
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4. In triangle PQR, Q = 90°, cos R = 0.6 and PQ = 8 cm. Find PR and RQ. (May draw the own diagram by above info provided)
Answer:
PR = 10
RQ = 6
Step-by-step explanation:
We have cos R = 0.6
also, cos R = adjacent/ hypotenuse
= RQ/PR
⇒ RQ/PR = 0.6
⇒ RQ = 0.6 PR -eq(1)
By pythagoras theorem,
PQ² + RQ² = PR²
given PQ = 8 and sub the value of RQ from eq(1):
8² + (0.6 PR)² = PR²
⇒ 64 + 0.36PR² = PR²
⇒ (1-0.36)PR² = 64
⇒ 0.64 PR² = 64
⇒ PR² = 64/0.64
⇒ PR² = 100
⇒ PR = 10
sub PR in eq(1):
RQ = 0.6*10
⇒ RQ = 6
. Calculez P(5) sachant que P(x) = x3 − 5x2 − 2x + 7.
Answer:
P(5) = - 3
Step-by-step explanation:
to evaluate P(5) substitute x = 5 into P(x)
P(5) = (5)³ - 5(5)² - 2(5) + 7
= 125 - 5(25) - 10 + 7
= 125 - 125 - 3
= 0 - 3
= - 3
A jewelry company makes copper heart pendants. Each heart uses 0.75in® of copper and there is o.323 pound of copper per cubic inch. If copper costs $3.68 per pound, what is the total cost for 24 copper hearts?
The total cost for 24 copper hearts would be $21.41.
To calculate the total cost for 24 copper hearts, we need to determine the total amount of copper used and then multiply it by the cost of copper per pound.
First, let's find out the total amount of copper used for 24 copper hearts. Each heart uses 0.75 square inches of copper, so for 24 hearts, the total amount of copper used would be:
0.75 square inches/heart [tex]\times 24[/tex]hearts = 18 square inches.
Next, we need to convert the square inches into cubic inches. Since we don't have information about the thickness of the hearts, we'll assume they are flat hearts with a thickness of 1 inch. Therefore, the volume of copper used for the 24 hearts would be:
18 square inches [tex]\times 1[/tex] inch = 18 cubic inches.
Now, we can calculate the total weight of copper used. Given that there is 0.323 pounds of copper per cubic inch, the total weight of copper for the 24 hearts would be:
18 cubic inches [tex]\times 0.323[/tex] pounds/cubic inch = 5.814 pounds.
Finally, we multiply the total weight of copper by the cost of copper per pound to find the total cost:
5.814 pounds [tex]\times[/tex] $3.68/pound = $21.41.
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A basket of cucumbers contains 10 cucumbers that were grown using conventional methods and 22 cucumbers that were grown using organic methods. If a customer randomly selects 5 cucumbers, what is the probability they select two conventional cucumbers and 3 organic cucumbers?
Use the definition of inverses to determine whether f and g are inverses.
Are the given functions inverse: A. No.
What is an inverse function?In Mathematics and Geometry, an inverse function is a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).
In order to determine whether f(x) and g(x) are inverses, we would determine the corresponding composite function of f(x) and g(x) in simplified form as follows;
(fog)(x) = -5[-1/5(x) - 9] + 9
(fog)(x) = x + 45 + 9
(fog)(x) = x + 54
(gof)(x) = -1/5(-5x + 9) - 9
(gof)(x) = x - 9/5 - 9
(gof)(x) = x - 54/5
Since (fog)(x) and (gof)(x) are not equal to x, we can conclude that f(x) and g(x) are not inverses of each other.
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I need help please!!
Answer:
(r q)(-3) = -3
(q r)(-3) = -3
Step-by-step explanation:
let x = 1
q(1) = -1 +2 = 1
r(1) = 1² = 1
(r q)(-3) = ?
(1×1)(-3) = -3
(q r)(-3) = ?
(1×1)(-3) = -3
Suppose that an object is thrown upward from ground level with an initial velocity of 160ft/sec. Its height after t seconds is a function h given by h(t)=-16t^2 +160t.
a) Find an equivalent expression for h(t) by factoring out a common factor with a negative coefficient.
b) Check your factoring by evaluating both expressions for h(t) at t=1.
The factored expression is
a) The factored expression for h(t) is -16t(t - 10), obtained by factoring out a common factor of -16 and a common factor of t from the original expression -16t^2 + 160t.
b) Both the original expression -16t^2 + 160t and the factored expression -16t(t - 10) yield the same result of 144 when evaluated at t = 1, confirming the correctness of the factoring.
a) To factor out a common factor with a negative coefficient from the expression h(t) = [tex]-16t^2 + 160t[/tex], we can rewrite it as:
h(t) = [tex]-16(t^2 - 10t)[/tex]
Now, let's focus on factoring the quadratic expression inside the parentheses. We can factor out a common factor of t:
h(t) = -16t(t - 10)
Therefore, the factored expression for h(t) is -16t(t - 10).
b) To check the factoring by evaluating both expressions for h(t) at t = 1, we substitute t = 1 into the original expression and the factored expression and compare the results.
Using the original expression:
h(1) = [tex]-16(1)^2 + 160(1)[/tex]
h(1) = -16 + 160
h(1) = 144
Using the factored expression:
h(1) = -16(1)(1 - 10)
h(1) = -16(1)(-9)
h(1) = 144
Both expressions yield the same result of 144 when evaluated at t = 1. Therefore, the factoring is correct.
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A jar of kosher dill spears is filled to the brim with a vinegar based pickling liquid and then
sealed. The base of the cylindrical jar has an area of 45 cm² and the height of the jar is
13 cm. When the pickles are opened, all the pickle juice is drained into a measuring cup,
amounting to 160 cm³ of pickle juice. Find the total volume of the dill spears.
The total volume of the dill spears is approximately 1013 cm³.
To find the total volume of the dill spears, we can use the formula for the volume of a cylinder, which is given by V = πr²h,
where V is the volume, r is the radius of the base, and h is the height of the cylinder.
First, let's find the radius of the base.
Since the area of the base is given as 45 cm², we can use the formula for the area of a circle,
A = πr², to solve for the radius.
Rearranging the formula, we have r = √(A/π).
Given that the area of the base is 45 cm², we can substitute this value into the formula to find the radius:
r = √(45/π) ≈ 3 cm (rounded to the nearest centimeter)
Now that we have the radius and the height of the jar, we can calculate the volume of the jar using the formula V = πr²h:
V = π(3²)(13) ≈ 1173 cm³ (rounded to the nearest cubic centimeter)
However, we need to subtract the volume of the pickle juice that was drained from the jar.
We are given that the amount of pickle juice is 160 cm³, so the total volume of the dill spears is:
Total volume = Volume of jar - Volume of pickle juice = 1173 cm³ - 160 cm³ = 1013 cm³
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