The area of the triangle with points Q(-1,1),R(2,-3), and S(-1,-3) is equal to 6 square units.
In Mathematics and Geometry, the area of a triangle can be calculated by using the following mathematical equation (formula):
Area of triangle, A = 1/2 × b × h
Where:
b represent the base area.h represent the height.By substituting the given vertices into the formula for the area of a triangle with coordinates, we have the following;
[tex]A=\frac{1}{2} \times |x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|\\\\A=\frac{1}{2} \times |-1(-3 + 3) + 2(-3 - 1) + (-1)(1 +3)|\\\\A=\frac{1}{2} \times |0 -8 - 4|\\\\A=\frac{1}{2} \times |-12|[/tex]
Area of triangle = 1/2 × 12
Area of triangle = 6 square units.
The population of City A was approximately 243000 people, and it increased by 8% in one year. What was the new population
Answer:
262,440
Step-by-step explanation:
Multiply the population (243000) times 8% (0.08)
243000*0.08 = 19,440
Add the 8% to the total population
19,440 + 243,000 = 262,440
The new population of City A after one year with an 8% of the increasing rate will be 262,440.
What is the percentage?The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.
The percentage is given as,
Percentage (P) = [Final value - Initial value] / Initial value x 100
The number of inhabitants in City A was roughly 243000 individuals, and it expanded by 8% in one year.
The new population of City A after one year with an 8% of the increasing rate is given as,
⇒ (1 + 0.08) x 243,000
⇒ 1.08 x 243,000
⇒ 262,440
The new population of City A after one year with an 8% of the increasing rate will be 262,440.
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Dam has been asked to post a parcel for a family member. He has been told that the parcel weighs 12 lbs 8 oz.
Answer:
The weight of the parcel is approximately 6 kilograms.
Step-by-step explanation:
The weight of the parcel is 12 lbs and 8 oz.
1 lb = 16 oz
So, 8 oz = lb.
That means, 12 lbs and 8 oz = (12 + 0.5) lbs = 12.5 lbs
Now, 1 lb = 454 g
So, 12.5 lbs = (12.5 × 454) g = 5675 g
As, 1 kilogram = 1000 g
That means, 1 g = 0.001 kilogram
So, 5675 g kilograms. (Rounded to the nearest whole kilogram)
Thus, the weight of the parcel is approximately 6 kilograms.
The base and the height of a parallelogram are multiplied by 6. Which of the following describes the effect of this change on the perimeter?
The perimeter is multiplied by 1/36.
The perimeter is multiplied by 1/6.
The perimeter is multiplied by 6.
The perimeter is multiplied by 36.
Answer:
The perimeter would also be multiplied by 6.
Step-by-step explanation:
The perimeter would also be multiplied by 6. Suppose the sides are 5 and 10. The perimeter would be the sum of the side lengths or 5 + 5 + 10 + 10 = 30.
Multiply each side length so 5*6 = 30 and 10*6 = 60. Find the perimeter by finding their sum, 30 + 30 + 60 + 60 = 180.
Divide the new perimeter by the old perimeter to find a scale factor.
180/30 = 6.
The new perimeter is exactly 6 times bigger since 6*30 = 180.
Answer:
The perimeter is multiplied by 6.
Step-by-step explanation:
can someone help me with this? Would greatly appreciate it ...
This is the answer to your mathematical question !!
Given that (x, y) = (5, 10), find r.
a.2√5
b.5√2
c.5√5
d.5√10
I am considering the point you have given is point on a circumference of a circle and center of circle is (0,0) and we have to calculate radius
so radius = root((x-h)^2 + (y-k)^2)
where (h,k) is center of circle
so r = root((5-0)^2+(10-0)^2)
= √(100+25) = √125 = 5√5
so answer is option c
The distance r from the point (5, 10) to the origin is calculated using the distance formula, resulting in r = 5√5. So, the correct answer is option (c) 5√5.
The student's question involves finding the distance r from a point to the origin in a coordinate system, given a point (x, y). Given the point (5, 10), we can calculate r using the distance formula [tex]r = \( \sqrt{(x - x_1)^2 + (y - y_1)^2} \).[/tex]
We want to find the distance r from the origin [tex](x_1,y_1)[/tex] = (0, 0) to this point (x, y) = (5, 10).
Substituting x = 5 and y = 10 into the equation, we get:
[tex]r = \( \sqrt{(5 - 0)^2 + (10 - 0)^2} \)\\ = \( \sqrt{5^2 + 10^2} \)\\ = \( \sqrt{25 + 100} \)\\ = \( \sqrt{125} \)\\ r = 5\sqrt{5}.[/tex]
Therefore, the correct answer is option (c) [tex]5\sqrt{5}[/tex].
The length of the edges of the triangle are (3x-4) feet, (x2-1)feet, and (2x-15) feet. What's is the perimeter of the triangle if x=4
Answer:
40 feet
Step-by-step explanation:
The perimeter of a triangle is the distance around the triangle. It can be found by adding all the sides together. First, find the amount each side is by substituting x = 4 and simplifying.
3x - 4 = 3(4) - 4 = 12 - 4 = 8
x² -1 = (4)² - 1 = 16 - 1 = 15
2x² - 15 = 2(4)² - 15 = 32 - 15 = 17
Add the sides together, 8 + 15 + 17 = 40 feet
10% of the students in a school like the color purple. 162 students don't like the color purple. How many students are in the school?
Answer:
There are 162 students in the school.
Step-by-step explanation:
10% do not like purple. x is the number of students in the school. We have 162 students that do not like purple.
x*.10 = 162
Divide each side by .10
x = 162/.10
x =1620
There are 162 students in the school.
The diameter of Jim's circular flower bed is 10 feet what is the area in square feet of Jim's flower bed?
Answer:
A = 25π ft² or A = 78.5398 ft²
Step-by-step explanation:
The formula for area of a circle is
A = πr² where r is the radius
We are given a diameter of 10, the radius is half of the diameter, so for our problem r = 5. Plug that in and evaluate
A = π(5²)
A = 25π
or
A = 78.5398 if you multiply out pi
A random experiment has three steps with three outcomes possible for the first step, two outcomes possible for the second step, and four outcomes possible for the third step. how many experimental outcomes exist for the entire experiment?
Answer: 6 outcomes
Step-by-step explanation: Since the first outcome is 2 and the second is 4 2 times 3 is ix since it is multiplying by 2 (2,4,6,8)
Final answer:
To find the total number of experimental outcomes for the entire experiment, multiply the number of outcomes for each step together, resulting in 24 total experimental outcomes.
Explanation:
The total number of experimental outcomes for the entire experiment can be calculated by multiplying the number of outcomes for each step together:
Number of outcomes for the first step: 3
Number of outcomes for the second step: 2
Number of outcomes for the third step: 4
Multiplying these numbers: 3 outcomes x 2 outcomes x 4 outcomes = 24 total experimental outcomes.
Add both the numerators and the denominators when adding fractions and mixed numbers. True False
Answer:
false
Step-by-step explanation:
Add both the numerators and the denominators when adding fractions and mixed numbers - this statement is false.
How to add numerators and denominators adding fractions and mixed numbers:
To add fractions with denominators, first find a common denominator. The least common denominator is easiest to use. The least common multiple can be used as the least common denominator. Adding mixed numbers involves adding the fractional parts, adding the whole numbers, and then recombining them as a mixed number.
1. Add the whole numbers together.
2. Find the lowest common denominator of both fractions.
3. Convert the fractions to have the LCD as their denominator.
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Complete the inequality statement.
14 ft ____ 4 1/2 yd.
Answer:
4.66667
Step-by-step explanation:
Help ASAP Please!!!!
Which function includes a translation of 3 units to the left?
A. f(x)= (x+1)^2-3
B. f(x)= (x+3)^2+1
C. f(x)= 3x^2+1
D. f(x)= (x-3)^2+1
Answer:
It's B. f(x) = (x + 3(^2 + 1.
Step-by-step explanation:
f(x) ---- > f(x + 3) is a translation of 3 units to the left.
The correct function that includes a 3 unit leftward translation is f(x) = [tex](x-3)^2+1.[/tex]
Explanation:The correct function that includes a translation of 3 units to the left is option D, f(x) = [tex](x-3)^2+1.[/tex]
When we translate a function to the left by a certain number of units, we subtract that number from the x-values. In this case, subtracting 3 units from x would result in a translation to the left. The function f(x) = [tex](x-3)^2+1.[/tex] represents this translation. The x-3 part shifts the graph horizontally to the left by 3 units.
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How are slopes and y-intercepts related to the number of solutions of a system of linear equations
For a system of 2 equations in 2 unknowns, there are 3 cases:
slopes are different — one solutionslopes are the same and y-intercepts are different — no solutionsslopes and y-intercepts are the same — infinitely many solutionsWhen slopes are different, the two lines intersect at one point, the solution.
When slopes are the same, the lines may be either parallel (different y-intercepts) or the same (same y-intercepts). If the lines are parallel, there are no points of intersection, hence no solutions. If the lines are the same line, they intersect at all points, so there are infinitely many solutions.
The corresponding polynomial function is f(x) = x2 + 5x - 1 f(x) = x2 + 4x - 5 f(x) = x2 + 5x - 5 f(x) = x2 - 4x - 5
Answer:
f(x)=x^2+4x-5
Step-by-step explanation:
Quadratic functions are second-order polynomials. To solve a quadratic equation, set the function equal to zero, and use the quadratic formula. The corresponding polynomial function in this case is f(x) = x² + 5x - 1.
Explanation:The Solution of Quadratic EquationsQuadratic functions are second-order polynomials. They are of the form f(x) = ax² + bx + c, where a, b, and c are constants. To solve a quadratic equation, set the function equal to zero, and use the quadratic formula: x = (-b ± √(b²-4ac))/(2a). The discriminant, b²-4ac, determines the number and nature of the solutions. If the discriminant is positive, there are two real solutions. If it is zero, there is one real solution (a perfect square). If it is negative, there are no real solutions, but there are two complex solutions.
In this case, the corresponding polynomial function is f(x) = x² + 5x - 1.
Julia bought 40 bags of compost each bag weighed 50 lb is how many tons of compost did she buy
Answer:
She bought .9 ton of compost.
Step-by-step explanation:
Take the amount of bags she bought and multiply it by the weight of each bag.
40*50=2,000
Then divide it by the amount a ton is.
2,000/2,204=0.90718474
Round it to the nearest tenth.
=.9
Then that's you're answer.
(Q6) What is the domain of the function f(x)= e^x/e^x+c if c is a constant greater than 0?
[tex]Domain: x\in\mathbb{R}[/tex]
Question 1(Multiple Choice Worth 2 points) Find the derivative of f(x) = 7 divided by x at x = 1.
-7
-1
1
7
Question 2(Multiple Choice Worth 2 points) Find the derivative of f(x) = 4x + 7 at x = 5.
4
1
5
7
Question 3(Multiple Choice Worth 2 points) Find the derivative of f(x) = 12x2 + 8x at x = 9.
256
-243
288
224
Question 4(Multiple Choice Worth 2 points) Find the derivative of f(x) = negative 11 divided by x at x = 9.
11/9
81/11
9/11
11/81
Question 5 (Essay Worth 2 points) The position of an object at time t is given by s(t) = 1 - 10t. Find the instantaneous velocity at t = 10 by finding the derivative.
1. Answer: a. -7
Step-by-step explanation:
[tex]f(x)=\dfrac{7}{x}\implies f(x)=7x^{-1}\\\\f'(x)=-1\cdot 7x^{-1-1}\\.\qquad =-7x^{-2}\\\\.\qquad =-\dfrac{7}{x^2}\\\\f'(1)=-\dfrac{7}{1^2}\\\\.\qquad =\large\boxed{-7}[/tex]
2. Answer: a. 4
Step-by-step explanation:
[tex]f(x)=4x+7\\f'(x)=1\cdot4x^{1-1}+0\cdot7\\.\qquad=4\\\\f'(5)=\large\boxed{4}[/tex]
3. Answer: d. 224
Step-by-step explanation:
[tex]f(x)=12x^2+8x\\f'(x)=2\cdot12x^{2-1}+1\cdot8x^{1-1}\\.\qquad=24x+8\\\\f'(9)=24(9)+8\\.\qquad=216+8\\.\qquad=\large\boxed{224}[/tex]
4. Answer: d. 11/81
Step-by-step explanation:
[tex]f(x)=\dfrac{-11}{x}\implies f(x)=-11x^{-1}\\\\f'(x)=-1\cdot -11x^{-1-1}\\\\.\qquad=11x^{-2}\\\\.\qquad=\dfrac{11}{x^2}\\\\f'(9)=\dfrac{11}{9^2}\\\\.\qquad=\large\boxed{\dfrac{11}{81}}[/tex]
5. Answer: -10
Step-by-step explanation:
s(t) = 1 - 10t
v = s'(t) = -10
s'(10) = -10
Please answer this question only if you know the answer!!! :)
A full circle is 360 degrees.
The circle is separated into 8 parts, so each arc is 1/8 of a circle, which equals: 360/8 = 45 degrees.
There are 2 45 degree arcs between B and C, so the arc from B to C = 45 x 2 = 90 degrees.
The answer is B.
According to a survey, 60 % of the residents of a city oppose a downtown casino. Of these 60 % about 8 out of 10 strongly oppose the casino. Complete parts (a) through (c). (a) Find the probability that a randomly selected resident opposes the casino and strongly opposes the casino. (b) Find the probability that a randomly selected resident who opposes the casino does not strongly oppose the casino. (c) Would it be unusual for a randomly selected resident to oppose the casino and strongly oppose the casino? Explain.
Answer:
(a) 0.48
(b) 0.20
(c) it is not unusual for a radomly selected resident to oppose the casino and strongly oppose the casino.
Step-by-step explanation:
(a) Find the probability that a randomly selected resident opposes the casino and strongly opposes the casino.
The probability that a radomly selected resident opposes the casino and strongly opposes the cassino is the product of the two probabilities, that a resident opposes the casino and that it strongly opposes the casino (once it is in the first group) as it is shown below.
Use this notation:
Probability that a radomly selected resident opposes the casino: P(A)Probability that a resident who opposes the casino strongly opposes it: P(B/A), because it is the probability of event B given the event Ai) Determine the probability that a radomly selected resident opposes the casino, P(A)
Probability = number of favorable outcomes / number of possible outcomes
P(A) is given as 60%, which in decimal form is 0.60ii) Next, determine,the probability that a resident who opposes the casino strongly opposes it, P(B/A):
It is given as 8 out of 10 ⇒ P(B/A) = 8/10iii) You want the probability of both events, which is the joint probability or intersection: P(A∩B).
So, you can use the definition of conditional probability:
P(B/A) = P(A∩B) / P(A)iv) From which you can solve for P(A∩B)
P(A∩B) = P(B/A)×P(A) = (8/10)×(0.60) = 0.48(b) Find the probability that a randomly selected resident who opposes the casino does not strongly oppose the casino.
In this case, you just want the complement of the probability that a radomly selected resident who opposes the casino does strongly oppose the casino, which is 1 - P(B/A) = 1 - 8/10 = 1 - 0.8 = 0.2.
(c) Would it be unusual for a randomly selected resident to oppose the casino and strongly oppose the casino?
You are being asked about the joint probability (PA∩B), which you found in the part (a) and it is 0.48.
That is almost 0.50 or half of the population, so you conclude it is not unusual for a radomly selected resident to oppose the casino and strongly oppose the casino.
What is the area of a triangle with a=25 b=13 and c=17? (picture provided)
For this case we must find the area of a scalene triangle (different sides) by means of Heron's formula:
[tex]S = \sqrt {p (p-a) (p-b) (p-c)}[/tex]
Where:
S: It's the area
p: It is the semiperimeter
a, b, c: They are the sides:
[tex]p = \frac {a + b + c} {2}\\p = \frac {25 + 13 + 17} {2}\\p = 27.5 \ units[/tex]
So:
[tex]S = \sqrt {27.5 (27.5-25) (27.5-13) (27.5-17)}\\S = \sqrt {10467.1875}\\S = 102.309 \ units ^ 2[/tex]
ANswer:
Option D
What facts are true for the graph of the function listed below? Please check all that apply.
F(x)=2 x 5^x
Answer:
E, B, A,
Step-by-step explanation:
C is wrong because it isn't decreasing
D is wrong because if X=0 the 5 to the power of zero would be 1 and 2*1= 2 meaning y goes lower than 5
F is wrong because if X=0 the 5 to the power of zero would be 1 and 2*1= 2 meaning the y intercept is 2 not 5
Answer:
[tex]\Large \boxed{\mathrm{A , \ B, \ and \ E}}[/tex]
Step-by-step explanation:
[tex]F(x)=2 \cdot 5^x[/tex]
There are no restrictions on x. The domain is all real numbers.
To find the y-intercept, we set the x value to 0. The y-intercept of the function is (0, 2).
The function is increasing, because as the value of the input increases, the value of the output increases as well.
Max wants to buy a new DVD player for $89.75. If sales tax is 8% how much will he pay in total for the DVD player
Answer:
$96.93
Step-by-step explanation:
1. multipy the cost of the item (89.75) times the 8% tax.
89.75 * 0.08 = 7.18 in tax
2. add the item cost plus tax
89.75 + 7.18 = $96.93
Max will have to pay $ 96.93 in total for the DVD player.
What is sales tax?An extra amount of money to be paid to the seller as tax on sales or on the receipt of sales is called sales tax.
How to find how much will he pay in total for the DVD player?According to the problem,
Max wants to buy a new DVD player for $89.75The sales tax is 8%So, Max will have to pay $ { 89.75 + [tex](\frac{8}{100}) 89.75[/tex]}
= $ 96.93
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Simplify the irrational number 128; then estimate it to two decimal places.
The square root of 128, which is an irrational number, is approximately 11.304. Rounded to two decimal places, it becomes 11.30.
Explanation:The process of simplifying the irrational number 128 involves finding a square root since irrational numbers are those that cannot be expressed as a simple fraction, such as roots that are not even. The square root of 128 is not a whole number, therefore it's an irrational number. Using a calculator, the square root of 128 is approximately 11.304. So to obtain the square root of 128 to two decimal places, you round 11.304 to 11.30.
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Find the inverse of the matrix [tex]\left[\begin{array}{cc}-4&6\\8&-12\\\end{array}\right][/tex]
if it exist.
Answer:
D
Step-by-step explanation:
If there is a 2x2 matrix as [tex]\left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right][/tex]
The determinant is given by [tex]\frac{1}{ad-bc}\left[\begin{array}{ccc}d&-b\\-c&a\\\end{array}\right][/tex]
Now, if we calculate the value of "ad - bc", we see that:
[tex](-4)(-12)-(8)(6)=0[/tex]
We can't calculate the inverse, the inverse doesn't exist. The answer is D.
Need Help ASAP!!.
1. Error Analysis: Anita Help drew the triangle below. Her friend, Greta Life, told her that her triangle was incorrect. Who is correct? Explain.
2. The Geo Air pilot is looking at SCCA from the plane. From the aircraft the angle of depression is 17 degrees. If the plane is at an altitude of 10,000 feet, approximately how far is the plane to SCCA? Round your answer to the nearest tenth. The image is not drawn to scale.
3. Trey Rigg’s house is next to a large pine tree. The National Weather Center has issued a warning for high winds. Trey is worried that his house is at risk of being crushed by the pine tree. Trey’s house is 25 feet from the base of the tree and Trey calculates the angle of elevation to be 43 degrees from the base of his house. Find the height of the tree to determine if Trey’s house is at risk. Round your answer to the nearest tenth.
Picture 1 is for Number 1.
Picture 2 is for Number 2.
Picture 3 is for Number 3.
Answer:
Part 1) Greta's right, the triangle is incorrect.
Part 2) [tex]34,203\ ft[/tex]
Part 3) The height of the tree is [tex]23.3\ ft[/tex], Trey’s house is not at risk
Step-by-step explanation:
Part 1) we know that
In the right triangle of the figure
[tex]sin(30\°)=\frac{1}{2}[/tex]
[tex]sin(30\°)=\frac{6\sqrt{3}}{12}=\frac{\sqrt{3}}{2}[/tex]
Compare
[tex]\frac{1}{2}\neq\frac{\sqrt{3}}{2}[/tex]
therefore
The triangle is not correct
Because
The side adjacent to the 30 degree angle should be [tex]6\sqrt{3}[/tex] and the side opposite the 30 degree angle should be [tex]6[/tex]
Part 2)
Let
x--------> the distance from the airplane to the SCCA (hypotenuse of the right triangle)
we know that
[tex]sin(17\°)=\frac{10,000}{x}[/tex]
[tex]x=\frac{10,000}{sin(17\°)}[/tex]
[tex]x=34,203\ ft[/tex]
Part 3)
Let
x--------> the height of the tree
we know that
[tex]tan(43\°)=\frac{h}{25}[/tex]
[tex]h=tan(43\°)(25)=23.3\ ft[/tex]
[tex]23.3\ ft< 25\ ft[/tex]
therefore
Trey’s house is not at risk
Solve the triangle (picture provided)
Answer:
m∠A = 9.1°
m∠B = 10.9°
c = 54.2
The answer is (d)
Step-by-step explanation:
∵ a = 25
∵ b = 30
∵ m∠C = 160°
∴ c² = a² + b² - 2 × ab × cos(C)
∴ c² = 25² + 30² - 2(25)(30) × cos(160) = 2934.5389
∴ c = 54.2
∵ a/sin(A) = b/sin(B) = c/sin(C)
∵ 25/sin(A) = 54.2/sin(160)
∴ sin(A) = (25 × sin(160)) ÷ 54.2 = 0.1577584
∴ m∠A = 9.1°
∵ 30/sin(B) = 54.2/sin(160)
∴ sin(B) = (30 × sin(160)) ÷ 54.2 = 0.18931
∴ m∠B = 10.9°
Find the value of a cylinder with a diameter of 8 inches and a height that is three times the radius
Answer:
nevermind
Step-by-step explanation:
Under which operations are polynomials closed? Addition, subtraction, multiplication, and division addition and subtraction only addition, subtraction, and multiplication only addition and multiplication only
Answer:
addition, subtraction, and multiplication only
Step-by-step explanation:
There are a couple of reasons why polynomials are not closed under division:
1. the set of polynomials includes zero, and division by zero is undefined.
2. division by a polynomial can give rise to terms that are not non-negative powers of the variable, 1/x, for example. These terms are not included in the set of polynomials.
Answer:
Addition, Subtraction, and Multiplication
Step-by-step explanation:
A trianglular prism with bases that are right triangles measuring 7 inches by 24 inches by 25 inches. The height of the prism is 3 inches
To calculate the volume of the triangular prism, first find the area of the right triangle that forms the base and then multiply it by the height of the prism.
Explanation:The question involves finding the volume of a triangular prism whose base is a right triangle with sides measuring 7 inches, 24 inches, and 25 inches, where 25 inches is the hypotenuse. The height of the prism is given as 3 inches. To find the volume of the prism, we use the formula for the volume of a triangular prism, which is the area of the triangular base times the height of the prism.
First, we find the area of the triangular base using the formula for the area of a right triangle, which is (1/2) × base × height. For this triangle, the base and height are the perpendicular sides, which are 7 inches and 24 inches respectively. Thus, the area of the base triangle is (1/2) × 7 × 24 square inches.
Next, we multiply the area of the base by the height of the prism to get the volume. The height of the prism is 3 inches, so the volume will be the area of the base triangle times 3 inches.
There are only a few roller coasters in the world whose path takes them underground. Suppose a portion of the roller coaster can be modeled by the function f(t) = (t^2 − 13t + 40)(t^2 − 2t + 3), where t represents the time elapsed in seconds since passing point A on the roller coaster and f(t) represents the height in feet above the ground.
Find the zeros of the function. What do these zeros represent in the context of the problem?
The thing is, I found the zeros ( which are [8,0] and [5,0]) but I don't know what they represent..
Answer: t=5 , t=8 on PLATO
Step-by-step explanation:
In the context of the problem, the zeros of the function represent the points in time when the roller coaster is at ground level. Specifically, the roller coaster goes underground at t = 5 seconds and comes back above ground at t = 8 seconds.
Explanation:
In the function f(t) = (t^2 − 13t + 40)(t^2 − 2t + 3), which models the height of the roller coaster above the ground, the zeros represent the points in time when the roller coaster is at ground level. In other words, when f(t) = 0, the height of the roller coaster is zero, implying it is at ground level. In your case, the zeros you found signify that the coaster is on the ground at the time instances t = 5 and t = 8 seconds since passing point A.
The values t = 5 and t = 8 seconds are the time instances when the roller coaster enters and exits the underground portion of the track respectively. It's crucial to comprehend the physical significance of mathematical solutions, particularly in real-world scenarios.
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