Considering they left at precise noon, at 12:45 they will be 9 miles apart after they leave the intersection.
Let's call the time it takes for them to be 9 miles apart "t". During this time, the truck will have traveled a distance of 30t, and the car will have traveled a distance of 42t. Since they are traveling in the same direction, we can subtract the distances to find the distance between them:
42t - 30t = 12t
So they are separating at a rate of 12 miles per hour.
To find when they will be 9 miles apart, we can set up the equation:
12t = 9
Solving for t:
t = 9/12
t = 0.75 hours,
To convert 0.75 hours into minutes, we have to multiply it by 60.
= 0.75 × 60
= 45 minutes
So they will be 9 miles apart 45 minutes after they leave the intersection, or at 12:45pm.
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Olga is a park ranger in a large national park. The
park is in a rural area that has no cellular phone
service. Olga has a walkie-talkie that allows her
to her to transmit weather updates to other
rangers in the park. Her walkie-talkie has a range
of 10 miles in all directions. The other park
rangers have receivers in their cars that can pick
up her weather updates when they are in range
of her walkie-talkie. On a particular day, Olga is at
the ranger station in the center of the park. Her
coworker Neil is doing a survey of the land by
driving along a straight road through the park.
The road that Neil is on starts 15 miles west and
15 miles south of the ranger station and ends 20
miles north and 10 miles east of the ranger
station.
For approximately how many miles of road will
Neil be able to receive Olga's weather updates?
Quadratic Equation _____?
Neil will be able to receive Olga's weather updates for a distance of 10 miles on the road.
What is equation?Equation is a mathematical statement that expresses the equality of two expressions. It is usually written using symbols, such as '=', '<', '>', '+', and '-'. Equations can be used to describe, model, and solve real-world problems. They can also be used to predict the behavior of a system or to find unknown values.
To calculate the approximate number of miles Neil will be able to receive Olga's weather updates, a quadratic equation can be used. The equation is as follows:
A = (x - 15)²+ (y - 15)² <= 100
where A is the area of a circle with a radius of 10 miles, centered at the ranger station, and (x,y) is the position of Neil's car on the road. The equation represents the area that is within the range of Olga's walkie-talkie, which is the area where Neil's car can receive Olga's weather updates.
To solve for x and y, we can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is equal to the square root of (x2 - x1)² + (y2 - y1)². Since we know the starting and ending points of Neil's road, we can use this formula to solve for x and y:
x = Square root of [ (10 - 15)² + (20 - 15)² ] = 10
y = Square root of [ (10 - 15)² + (10 - 15)² ] = 5
Substituting these values into the equation above, we get A = (10 - 15)² + (5 - 15)²= 100, which is Olga's walkie-talkie range. Therefore, Neil will be able to receive Olga's weather updates for a distance of 10 miles on the road.
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find the probability that the first arrival will not occur until at least the fifth one-second interval.
The probability that the first arrival will not occur until at least the fifth one-second interval is 1/32.
This is calculated by the following equation:
P(First arrival in at least the fifth one-second interval) = 1 - P(First arrival in first 4 one-second intervals)
P(First arrival in first 4 one-second intervals) = 4/32
Therefore, P(First arrival in at least the fifth one-second interval) = 1 - 4/32 = 1/32
Therefore, the probability that the first arrival will not occur until at least the fifth one-second interval is 1/32.
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an order of award presentations has been devised for seven people: jeff, karen, lyle, maria, norm, olivia, and paul. in how many different orders can the awards be presented so that maria and olivia will be one after the other?
There are 2 x 120 = 240 possible arrangements in which Maria and Olivia are adjacent.
An order of award presentations has been devised for seven people, including Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. We must find out how many different orders the awards can be presented in if Maria and Olivia will be one after the other.There are six possible pairs:
{M, O}, {O, M}, {J, M}, {M, K}, {L, M}, and {M, N}.
In the total number of arrangements, these pairs can occur in two different ways. They are:-
The pair can come first, and the remaining people can be arranged in the rest of the arrangement- The pair can come last, and the remaining people can be arranged in the rest of the arrangement.That is, each pair may appear at the beginning or end of the arrangement, and the remaining persons may be arranged in the remaining positions in any order.
The arrangements would be the same for all six pairs, so we can choose any pair and multiply the outcome by 6.Choose the pair {M, O}. The pair may appear at the start or end of the sequence. There are 2 ways to pick which pair appears first or second.There are 5 individuals left to be arranged. The remaining 5 people can be arranged in 5! = 120 ways.
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Carla asked students at a lunch table what main course they like. Out of those students,
28 like pizza, 15 like chicken nuggets and 8 like both. What is the probability that a
randomly selected student will like pizza but not chicken nuggets?
A. 4/5
B. 4/7
C. 15/28
D.8/35
Answer:
28/51
Step-by-step explanation:
First you add up all the values: 28 + 15 + 8 = 51. That is your denominator because the total is the denominator. Then you put 28 as your numerator because that's how many off all the people that only like pizza.
A’(10, 5) is the image of A after a translation along the vector 〈−6, 0〉. What are the coordinates of A?
To perform the opposite translation, we add the opposite of the translation vector to the image point A': the coordinates of point A are (16, 5).
what is a vector?
In mathematics, a vector is an object that represents a quantity having both magnitude (or length) and direction. Vectors can be represented geometrically as arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.
To find the coordinates of point A, we need to perform the opposite translation of moving along the vector 〈−6, 0〉 from the image point A'(10, 5). This is because a translation is a rigid motion that preserves the distance between points, so the distance between A and A' is the same as the distance between their respective translations.
To perform the opposite translation, we add the opposite of the translation vector to the image point A':
A = A' - 〈-6, 0〉 = (10, 5) - (-6, 0) = (16, 5)
Therefore, the coordinates of point A are (16, 5).
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Which expression is equivalent to Cube root of 343 x Superscript 9 Baseline y Superscript 12 Baseline z Superscript 6?
7x3y4z2
7x3y6z2
49x3y6z2
49x3y4z2
The value of the cube root [tex]343x^9y^{12}z^6[/tex] is [tex]7x^3y^4z^2[/tex]
A number's cube root is a quantity that when multiplied by itself three or more times, returns the original quantity. The cube root of 27, represented as 327, for instance, is 3 because 3 x 3 x 3 = 27 = 33 when we multiply 3 by itself three times. As a result, we can state that the cube root yields a value that is essentially cubed. Thus, 27 is referred to as the ideal cube. We may deduce what the cube's root is from the word "cube root." It denotes the integer that brought about the cube under the root.
The cube root 0f [tex]343x^9y^{12}z^6[/tex]
[tex]\sqrt[3]{343x^9y^{12}z^6} \\\sqrt[3]{(7x^3y^4z^2)^3} \\\\\\7x^3y^4z^2[/tex]
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Inga is solving 2x2 + 12x – 3 = 0. Which steps could she use to solve the quadratic equation? Select three options.
2(x2 + 6x + 9) = 3 + 18
2(x2 + 6x) = –3
2(x2 + 6x) = 3
x + 3 = Plus or minus sqrt of 21/2
2(x2 + 6x + 9) = –3 + 9
The steps that she could use to solve the quadratic equation is determined as 2(x² + 6x + 9) = 18 + 3.
Solution to the quadratic equationThe solution to the quadratic equation is determined as follows;
2x² + 12x - 3 = 0
2x² + 12x = 3
divide through by 2
x² + 6x = ³/₂
take half of coefficient of x, square it and add it both sides of the equation
(x + 3)² = ³/₂ + 3²
x² + 6x + 9 = 9 + ³/₂
2(x² + 6x + 9) = 18 + 3
Thus, the steps that she could use to solve the quadratic equation is determined as 2(x² + 6x + 9) = 18 + 3.
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The roots of the given quadratic functions are -
x = (- 6 ± √42)/2.
What is function?A function is a relation between a dependent and independent variable. We can write the examples of functions as -
y = f(x) = ax + b
y = f(x, y, z) = ax + by + cz
Given is that Inga is solving the quadratic equation -
2x² + 12x - 3 = 0
The quadratic equation is given as -
2x² + 12x - 3 = 0
We can writ its solution as -
x = {- 12 ± √(144 + 24)}/4
x = (- 12 ± √168)/4
x = (- 12 ± 2√42)/4
x = (- 6 ± √42)/2
Therefore, the roots of the given quadratic functions are -
x = (- 6 ± √42)/2.
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12.8 x 3/4 also One more is
One-fifth the sum of one-half and one-third this one u have to write in equivalent expression
Answer:
\
Step-by-step explanation:
Given expression: 3x+9
Take 3 outside from the expression, we get,
= 3(x+3), which is called the equivalent expresion
A bag of candy contained a total of 1,000 candies before it was opened. Four students each pulled out 20 candies at random and recorded the color of the candies in this table
To determine the most likely distribution of the colors of candies in the bag before it was opened, we can analyze the data from the table and try to find a distribution that would result in the observed frequencies.
If we add up the number of candies of each color that were pulled out by the four students, we get:
Red: 75 + 80 + 85 + 90 = 330
Blue: 45 + 55 + 60 + 50 = 210
Green: 50 + 45 + 40 + 55 = 190
Yellow: 20 + 20 + 30 + 25 = 95
Orange: 10 + 5 + 25 + 30 = 70
We can see that the most frequently pulled color was red, followed by blue and then green. This suggests that the bag may have had more red candies than the other colors before it was opened.
Looking at the answer choices, the distribution in option B (300 red, 250 blue, 200 green, 100 yellow, and 150 orange) seems to match the observed frequencies the closest, as it has the highest number of red candies. Therefore, option B is the most likely distribution of the colors of candies in the bag before it was opened.
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Correct question:
A bag of candy contained a total of 1,000 candies before it was opened. Four students each pulled out 20 candies at random and recorded the color of the candies in this table.
According to this data, which was the most likely distribution of the colors of candies in the bag before it was opened?
A. 250 red, 250 blue, 250 green, 125 yellow, and 125 orange
B. 300 red, 250 blue, 200 green, 100 yellow, and 150 orange
C. 200 red, 200 blue, 200 green, 200 yellow, and 200 orange
D. 200 red, 200 blue, 200 green, 100 yellow, and 100 orange
a. Is there a value of x, for -3≤x≤2, such that g(x)= 0
b. Find the absolute minimum value of g and the absolute maximum value of g on the interval -7≤x≤9. Justify your answer.
For x = -2, g(-2) = 2(-2)^3 - 5(-2)^2 + 4(-2) - 1 = 0, so there is a value of x such that g(x) = 0 for -3 ≤ x ≤ 2.
The absolute minimum value of g on the interval -7 ≤ x ≤ 9 is -765, and the absolute maximum value of g on the interval is 1720.
How to Solve the Problem?a. To determine if there is a value of x such that g(x) = 0 for -3 ≤ x ≤ 2, we can plug in each value of x in the interval into the equation and see if we get 0.
g(x) = 2x^3 - 5x^2 + 4x - 1
For x = -3, g(-3) = 2(-3)^3 - 5(-3)^2 + 4(-3) - 1 = -55, which is not 0.
For x = -2, g(-2) = 2(-2)^3 - 5(-2)^2 + 4(-2) - 1 = 0, so there is a value of x such that g(x) = 0 for -3 ≤ x ≤ 2.
b. To find the absolute minimum and maximum values of g on the interval -7 ≤ x ≤ 9, we can use the Extreme Value Theorem, which states that a continuous function on a closed interval will have both an absolute minimum and maximum value on that interval.
To find these values, we can take the derivative of g(x) and set it equal to 0 to find critical points, and then evaluate g(x) at those critical points as well as at the endpoints of the interval.
g(x) = 2x^3 - 5x^2 + 4x - 1
g'(x) = 6x^2 - 10x + 4 = 2(3x-2)(x-1)
Setting g'(x) = 0, we get critical points x = 2/3 and x = 1.
g(-7) = -765, g(2/3) = -23/27, g(1) = 0, and g(9) = 1720.
Therefore, the absolute minimum value of g on the interval -7 ≤ x ≤ 9 is -765, and the absolute maximum value of g on the interval is 1720.
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An angle measures 60° more than the measure of its supplementary angle. What is the measure of each angle?
Answer:
The angles are 60° and 120°.
Step-by-step explanation:
"supplementary" means that the two angles add up to 180°.
One angle can be x.
The other angle can be x+60° bc its 60° more.
Write an equation.
x + x + 60 = 180
combine like terms.
2x + 60 = 180
subtract 60
2x = 120
divide by 2
x = 60.
Since x = 60, one angle is 60° and the other is 120° (that's 60° more) and they are supplementary.
60° + 120° = 180°
The angles are 60° and 120°.
The supplement of an angle is 30 more than twice its complement. What is the measure of the
angle?
Answer: 30
180 - x = 180 - 2x + 30
x = 30
Answer:
The measure of the unknown angle is 30°.
Step-by-step explanation:
Let the measure of the unknown angle be x°.
Supplementary angles are two angles whose measures sum to 180°.
Complementary angles are two angles whose measures sum to 90°.
Therefore, the supplement of x° is (180 - x)°, and its complement is (90 - x)°.
Given that the supplement is 30° more than twice its complement:
(180 - x)° = 2(90 - x)° + 30°
To find the measure of the angle, solve the equation:
⇒ (180 - x)° = (180 - 2x)° + 30°
⇒ 180° - x° = 180° - 2x° + 30°
⇒ 180° - x° = 210° - 2x°
⇒ 180° - x° + 2x° = 210° - 2x° + 2x°
⇒ 180° + x° = 210°
⇒ 180° + x° - 180° = 210° - 180°
⇒ x° = 30°
Therefore, the measure of the unknown angle is 30°.
8. A $15,000 Honda depreciates at the rate of 12% per year.
a) Write an equation.
b) How much is the car worth in 5 years?
c) To the nearest year, when will the car be worth $1000?
According to the given information, the equation is [tex]V = 15,000(1-0.12)^t[/tex], the car is worth $7,156.67 in 5 years and will be worth $1000 in 24 years.
What is the depreciation rate?
Depreciation rate is a mathematical process in which a quantity decreases over time in a manner proportional to its current value. This means that the rate of decay is proportional to the amount of the substance remaining, and as the amount of the substance decreases, the rate of decline also decreases
a) V = P(1-r)^t, where V is the value of the car, P is the initial price ($15,000), r is the depreciation rate (0.12), and t is the time in years.
[tex]V = 15,000(1-0.12)^t[/tex]
b) To find the value of the car in 5 years, substitute t = 5 into the equation:
[tex]V = 15,000(1-0.12)^5[/tex]
V = 7,156.67
The car is worth approximately $7,156.67 in 5 years.
c) To find when the car will be worth $1000, we need to solve for t in the equation:
[tex]1000 = 15,000(1-0.12)^t[/tex]
[tex]0.067 = (1-0.12)^t[/tex]
Take the natural logarithm of both sides:
ln(0.067) = t ln(0.88)
t = ln(0.067) / ln(0.88)
t ≈ 23.7
Therefore, the car will be worth $1000 in approximately 24 years.
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This histogram shows the number of people who volunteered for community service and the number of hours they worked.
How many volunteers worked no more than 10 hours?
A 30 volunteers
B 35 volunteers
C 55 volunteers
D 60 volunteers
A 2-yard piece of rope costs $22. 86. What is the price per foot?
The price per foot of the 2-yard rope is $6.36.
To find the price per foot of the 2-yard rope, we need to first convert the length to feet. Since there are 3 feet in a yard, a 2-yard rope is equal to 6 feet.
Next, we divide the total cost of the rope ($22.86) by the length of the rope in feet (6 feet) to get the price per foot:
$22.86 ÷ 6 = $3.81 per yard
To convert this to the price per foot, we divide $3.81 by 3 (since there are 3 feet in a yard):
$3.81 ÷ 3 = $1.27 per foot
Therefore, the price per foot of the 2-yard rope is $1.27 x 5 = $6.36.
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If t represents an odd integer , which of the following expressions represents an even integer ?
A) t + 2
B) 2t - 1
C) 3t - 2
D) 3t + 2
E) 5t + 1
Answer: A) t + 2
Step-by-step explanation:
If t represents an odd integer, then it can be written as t = 2k + 1 for some integer k.
A) t + 2 = (2k + 1) + 2 = 2k + 3 = 2(k + 1) + 1, which is an odd integer.
B) 2t - 1 = 2(2k + 1) - 1 = 4k + 1, which is an odd integer.
C) 3t - 2 = 3(2k + 1) - 2 = 6k + 1, which is an odd integer.
D) 3t + 2 = 3(2k + 1) + 2 = 6k + 5 = 2(3k + 2) + 1, which is an odd integer.
E) 5t + 1 = 5(2k + 1) + 1 = 10k + 6 + 1 = 2(5k + 3) + 1, which is an odd integer.
Therefore, the only expression that represents an even integer is option A) t + 2.
a postal employee is counting how many houses on the route have dogs. statistically, 48\% of homes have dogs, according to the postal employees research. on the route of 30 homes, the postal employee encounters 12 dogs. what is the variance?
The variance is 7.2.
We can use the binomial distribution formula to calculate the variance. The formula for the variance of a binomial distribution is
Variance = n × p × (1 - p)
where n is the number of trials, p is the probability of success on each trial, and (1 - p) is the probability of failure on each trial.
In this case, n = 30 (the number of homes on the route), p = 0.48 (the probability of a home having a dog), and (1 - p) = 0.52 (the probability of a home not having a dog).
The postal employee encountered 12 dogs, which means there were 18 homes without dogs. So, the actual probability of success in this case is
P(dogs) = 12/30 = 0.4
Using the formula above, we can calculate the variance
Variance = n × p × (1 - p)
= 30 × 0.4 × 0.6
= 7.2
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i rlly dont understand this but i need the answer asap
Answer: 163.36 in
Step-by-step explanation: Using the formula C=2πr you can plug in the radius and get your answer.
C=2 x pi x 26
C = 163.362817987
round as needed
Answer: 163.36
Im pretty sure all you have to do is use the formula
C=2πr
Step-by-step explanation:
C=2π(26)
163.36
right cone has a base with diameter 16 units.
me volume of the cone is 320 cubic units.
What is the length of a segment drawn from the apex to the edge of the circular base?
Answer:
The length of the segment is 8 units.
We can use the formula for the volume of a cone to calculate the height of the cone:
V = (1/3)πr^2h
320 = (1/3)π(8^2)h
h = 320/(1/3)(π)(64)
h = 8 units
4 Applying torque A machine fastens plastic screw-on
caps onto containers of motor oil. If the machine
applies more torque than the cap can withstand, the
cap will break. Both the torque applied and the strength
of the caps vary. The capping-machine torque T follows
a Normal distribution with mean 7 inch-pounds and
standard deviation 0.9 inch-pound. The cap strength C (the torque that would break the cap) follows a normal distribution with mean 10 inch pounds and standard deviation 1.2 inch pounds
A find the probability that a randomly selected cap had a strength greater than 11 inch pounds
The probability that a randomly selected cap had a strength greater than 11 inch-pounds is approximately 0.2033 or 20.33% (rounded to two decimal places).
We know that cap strength, C, follows a normal distribution with mean μ = 10 inch-pounds and standard deviation σ = 1.2 inch-pounds. Therefore, we can standardize the variable C using the z-score formula:
z = (x - μ) / σ
where x is the value of cap strength we are interested in. In this case, we want to find the probability that a randomly selected cap had a strength greater than 11 inch-pounds, so x = 11. Plugging in the values for μ and σ, we get:
z = (11 - 10) / 1.2 = 0.83
Next, we need to find the probability that a standard normal random variable Z is greater than 0.83. This can be looked up in a standard normal distribution table or calculated using software. Using a standard normal distribution table, we find that the probability of Z being greater than 0.83 is approximately 0.2033.
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A dental student is conducting a study on the number of people who visit their dentist regularly. Of the 520 people surveyed, 312 indicated that they had visited their dentist within the past year.Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=60.Round all answers to 3 decimal places.p=Up=Op=
Answer:
Step-by-step explanation:
312/520 equals 60%
other people 40%
The price of a 6-minute phone call is 1. 80. What is the price of a 18-minute phone call?
Answer:
5.40
Step-by-step explanation:
let X represent the price of a 18 minute phone call
X=18 ×1.80
6
= 5.40
1. Divide and simplify. **Please show all work to receive full credit 1/x+4 ÷ x-3/ x²+7x+12
After dividing and simplifying we get (x-3)/(x+3)
What is division?One of the four fundamental operations of arithmetic, or how numbers are combined to create new numbers, is division. The additional operations are addition, subtraction, and multiplication.
At a fundamental level, counting the instances in which one number is contained within another is one interpretation of the division of two natural numbers. There is no requirement that this quantity be an integer. For instance, if there are 20 apples and they are divided equally among four people, each person will get 5 apples (see picture).
The integer quotient, which is the quantity of times the second number is entirely contained in the first number, and the remainder are both produced by the division with remainder or Euclidean division of two natural numbers.
1/(x+4) divide by [tex](x-3)/(x^2 + 7x +12)[/tex]
Simplify : [tex]x^2 +7x +12[/tex]
[tex]x^2 + (3+4)x + 12[/tex]
[tex]x^2 + 3x +4x + 12[/tex]
[tex]x(x + 3) + 4(x + 3)[/tex]
[tex](x+3)(x+4)[/tex]
[tex]1/(x + 4) x (x-3)/(x+3)(x-4)[/tex]
After dividing and simplifying we get the following answer;
[tex]= (x-3)/(x+3)[/tex]
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In circle R with the measure of minor arc
QS= 120°, find m/QTS.
Answer:180
Step-by-step explanation:
Find the circumference of each circle.Use your calculators value of pi.Round your answer to the nearest tenth.
Answer:
5) 59.7 in
6) 39.0 mi
7)
8) 50.3 mi
Step-by-step explanation:
5) C = [tex]\pi (19)[/tex]
C = 59.7 in
6) C = [tex]2\pi (6.2)[/tex]
C = 39.0 mi
7) RADIUS IS CUT OFF
use C = [tex]2\pi r[/tex]
8) C = [tex]2\pi (8)[/tex]
C = 50.3 mi
Bob works at Goodburger and gets a 20% discount. He wants to buy a burger that has a menu price of $4.75. What will his discount be?
Answer:
20÷100×4.75=0.95
4.75-0.95=$3.8
Answer:
i got 4.55$
Step-by-step explanation:
i just converted the percentage (20%) and then subtracted that number (0.2) from the original price (4.75$)
i really need help, my test is tomorrow and it's crucial to my grade. im offering 30 points please
Answer:
9, 10, 3y, 5, 3y, 5
Step-by-step explanation:
First look at the second row
_(2y+3)+_(2y+3)
We know that if we multiply the first blank time 2y is should equal 6y^2, so the answer for the first one is 3y. The third blank times 3 should be 15, so the answer for the third one is 5. Therefore, 3y(2y+3)+5(2y+3).
Now if we distribute, we get the answer for the first part: 6y^2+9y+10y+15.
In the second line we got, 3y(2y+3)+5(2y+3), we can factor out the 2y+3 and end up with (2y+3)(2y+5).
the ratio of purple skittles to orange skittles in the bag is 5: 2. if there are 112 skittles in the bag, how many of them are orange?
Answer is in the photo attached. Hope this helped!
The number of orange skittles in the bag is 32.
What is the ratio?Ratio is described as the comparison of two quantities to determine how many times one obtains the other. The proportion can be expressed as a fraction or as a sign: between two integers.
We are given that;
The ratio of purple skittles to orange skittles= 5:2
Total skittles= 112
Now,
Let's call the number of purple skittles in the bag "5x" and the number of orange skittles in the bag "2x", where x is a common factor.
We know that the total number of skittles in the bag is 112. So we can write:
5x + 2x = 112
Simplifying, we get:
7x = 112
Dividing both sides by 7, we get:
x = 16
Now we can find the number of orange skittles by substituting x = 16 into our expression for the number of orange skittles:
2x = 2(16) = 32
Therefore, by the given ratio answer will be 32 orange skittles in the bag.
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In circle P with the measure of arc NQ= 82°, find m/NPQ.
Answer:
m<NPQ= 82°
Step-by-step explanation:
Below is the explanation for the answer to the assignment.
a. Is there a value of x, for -3≤x≤2, such that g(x)= 0
b. Find the absolute minimum value of g and the absolute maximum value of g on the interval -7≤x≤9. Justify your answer.
a) There are three values of x, namely -3, -2, and -1, for which g(x) = 0.
b) The absolute minimum value of g on the interval -7 ≤ x ≤ 9 is 20, and the absolute maximum value of g is 90
How can we solve?a. To check if there is a value of x for -3 ≤ x ≤ 2 such that g(x) = 0, we can plug in each value in the interval and see if any of them make g(x) equal to 0:
g(-3) = -3² + 5(-3) + 6 = 0
g(-2) = -2² + 5(-2) + 6 = 0
g(-1) = -1² + 5(-1) + 6 = 0
g(0) = 0² + 5(0) + 6 = 6
g(1) = 1² + 5(1) + 6 = 12
g(2) = 2² + 5(2) + 6 = 20
So, we can see that there are three values of x, namely -3, -2, and -1, for which g(x) = 0.
b. To find the absolute minimum and maximum values of g on the interval -7 ≤ x ≤ 9, we can first find the critical points of g by taking its derivative:
g'(x) = -2x + 5
Setting g'(x) = 0, we get:
-2x + 5 = 0
-2x = -5
x = 5/2
So, the only critical point of g is x = 5/2.
We can now check the values of g at the endpoints of the interval and the critical point:
g(-7) = -7² + 5(-7) + 6 = 20
g(9) = 9² + 5(9) + 6 = 90
g(5/2) = (5/2)² + 5(5/2) + 6 = 43.25
Therefore, the absolute minimum value of g on the interval -7 ≤ x ≤ 9 is 20, which occurs at x = -7, and the absolute maximum value of g is 90, which occurs at x = 9.
We can justify this by noting that g is a quadratic function with a negative leading coefficient, which means that it opens downward and has a maximum value at its vertex (which occurs at x = 5/2). Since the vertex is within the given interval, and the function is decreasing on the left and increasing on the right of the vertex, the maximum and minimum values of g must occur at the endpoints of the interval.
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