The Pythagorean Theorem can be used to solve for the missing sides of a triangle when given two other sides
a. The missing components that can be solved using the given values are the missing sides of the triangle (side b and side c).
b. To find the missing side, we can use the Pythagorean Theorem, which states that a^2 + b^2 = c^2, where a and b are the two sides of the triangle, and c is the hypotenuse. Therefore, we can solve for c by taking the square root of (a^2 + b^2).
The Pythagorean Theorem can be used to solve for the missing sides of a triangle when given two other sides. To do this, we take the square root of the sum of the squares of the two given sides, which gives us the length of the hypotenuse.
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The function f is defined for all numbers x by f(x)=x^2+x. If t is a number such that f(2t)=30, which two of the following could be the number t ?
Indicate two such numbers.
−5
−3
−1/2
2
5/2
The two numbers which could be used for two in function are 5/2 and - 3.
What is function?The definition of a function is a "well-behaved" relation is one in which, given a starting point, we know the exact one ending spot to go to; given an x-value, we get only and precisely one corresponding y-value. Because functions pair information, they are all relations, but not all relations are functions. A subset of all of your relations are well-behaved family members, just as all mathematical relations are well-behaved functions.)
Given f(x)=x²+x
and f(2t)=30
When x = 2t, x²+x = 30
Substitute the value of x in function
(2t)²+2t = 30
4t² + 2t = 30
2(2t² + t) = 30
2t² + t = 15
2t² + 3t - 5t - 15 = 0
2t(t +3) - 5(t + 3)
(2t - 5)(t + 3)
t = 5/2
t = -3
Thus, The two numbers which could be used for two in function are 5/2 and - 3.
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Check whether
(
2
,
11
)
is a solution to the system, then choose all true statements from the list.
Equation 1:
−
2
x
+
5
y
=
34
Equation 2:
−
7
y
=
−
61
−
8
x
Group of answer choices
The point does not satisfy either equation
The point satisfies only equation 1
The point satisfies both equation 1 and 2
The point satisfies only equation 2
The point is a solution to the system
The point is not a solution to the system
The correct statements from the group of answer choices are:
The point satisfies both equation 1 and 2
The point is a solution to the system
What is linear equation ?
A linear equation is an algebraic equation with simply a constant and a first-order (linear) term of the form y=mx+b, where m is the slope and b is the y-intercept. The above is sometimes referred to as a "linear equation with two variables," where y and x are the variables.
To check whether the point (2, 11) is a solution to the system, we need to substitute these values into both equations and see if they hold true.
For equation 1:
-2 * 2 + 5 * 11 = -4 + 55 = 51
For equation 2:
-7 * 11 = -77
-61 - 8 * 2 = -61 - 16 = -77
Since both equations are true for the given point, the statement "The point satisfies both equation 1 and 2" is correct and "The point is a solution to the system" is also correct.
Therefore, the correct statements from the group of answer choices are:
The point satisfies both equation 1 and 2
The point is a solution to the system
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Roy took out a 30-year loan for $155,000 at an APR of 5. 5%, compounded
monthly. Approximately what would be the total cost of his loan if he paid it
off 4 years early?
O
A. $312,423. 90
O
B. $37,842. 06
O
O
c. $274,581. 84
D. $316,825. 20
Option A is the correct Answer. The total cost of his loan if he paid it off 4 years early is 312,423.90
We have given:
The principal amount which is 30,000
Time which is 30 year
The rate which is 5.5%
And since, we have to find 4 years early so, the time would be: 20-4=16 years.
And since, we have to find for 12 months
Hence, n=12
We have the formula to calculate compound interest:
[tex]A=P(1+\frac{r}{n})^n^t[/tex]
Where
A=compound interest
P= Orginal amount deposited.
r= rate
n=times its compounds per year monthly
t= years.
P=155000, r=5.5, t=16
On substituting the values we get:
[tex]A=155000(1+\frac{0.055}{12})^1^2^*^1^6\\\\A=155000(1+\frac{0.055}{12})^1^9^6\\\\A=155000(14884.0754)\\\\A=312,423. 90[/tex]
Therefore, the compound interest is 312,423.90
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Today, Thurber tudie math for 15 minute and he tudie cience for 75 minute. From now on, Thurber plan to tudy math for 50 minute per day and to tudy cience for 30 minute per day. Will Thurber ever have tudied the ame total amount of time for both math and cience? Explain. Ue the drop-down menu to complete the entence. 5
day from today, Thurber will have tudied
Chooe. Of math and
Chooe. Minute of cience. I need help jut pleae give me the anwer i feel my i light headed
After three days Thurber will have studied the same total amount of time for both math and science and the time is 165 minutes for both.
How do you find the time in math?Time can be described in mathematics as an ongoing and continuous series of events that take place one after another, from the past through the present, and into the future. The duration of events or the gaps between them can be measured, compared, or even ordered using time.The equation for calculating time is [Time = Distance ÷ Speed].Seconds, minutes, hours, days, weeks, months, and years are the fundamental units of time. To determine the time of day, we use seconds, minutes, and hours; to determine the date, we use days, months, and years. The smallest units of time are seconds, minutes, and hours, while the larger ones are days, months, and years.Given data :
Today, he studies 15 minutes math and 75 minutes science and from now on, 50 minutes math and 30 minutes science. So if he studies for 1 day total time for math and science will be:
Math = 15 + 50 = 65
Science = 75 + 30 = 105
2nd day:
Math = 65 + 50 = 115
Science = 105 + 30 = 135
3rd day:
Math = 115 + 50 = 165
Science = 135 + 30 = 165
So, after three days Thurber will have studied the same total amount of time for both math and science and the time is 165 minutes for both.
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Helpp! ASAP You'll earn 90 points! it's urgent
a shopper purchased 7 t shirts and 5 pairs of pants for $200. the next day, he purchased 5 t-shirts and one pair of pants for $112. what is a system of equations that represents the soulution? How much does each t-shirt and each pair of pants cost?
The system of equations are 7x + 5y = 200 and 5x + y = 112. The cost of each t shirt is $20 while the cost of each pants is $12.
What is an equation?An equation is an expression showing the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on depending on the degree.
Let x represent the cost of each t shirt and y represent the cost of each pants.
The shopper purchased 7 t shirts and 5 pairs of pants for $200, hence:
7x + 5y = 200 (1)
Also, he purchased 5 t-shirts and one pair of pants for $112, hence:
5x + y = 112 (2)
Solving the two equations simultaneously gives:
x = 20, y = 12
The system of equations are 7x + 5y = 200 and 5x + y = 112
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Ben collected data from a group of 12 people. He measured each person’s resting heart rate and recorded the average number of hours each person exercised per week. He created a scatterplot to show the data he collected.
Based on the scatterplot, what is the best prediction of the resting heart rate, in beats per minute, of a person who exercises an average of 8 hours each week?
Based on the scatterplot, 50 beats per minute is the best prediction of the resting heart rate, in beats per minute, of a person who exercises an average of 8 hours each week.
What is a scatter plot used for?In an effort to demonstrate the degree to which one variable is influenced by another, scatter plots are used to plot data points on a horizontal and vertical axis. The values of the columns set on the X and Y axes determine the position of the marker that represents each row in the data table.With a specific confidence range, a scatter plot can infer many types of correlations between variables. For instance, height would be on the x-axis while weight and height would be on the y-axis. Positive (increasing), negative (falling), or no correlations may exist (uncorrelated).The higher the correlation between the two variables, or the tighter the association, the more closely the data points resemble a straight line when they are plotted.Learn more about scatter plot refer to ;
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There are only green, yellow and black beads in a bag.
A bead is taken at random from the bag.
Complete the table below.
Colour
Green
Yellow
Black
Number of beads
8
8
Probability
15
Green Probability: 2/5
Yellow Probability: 2/5
Black Number of Beads: 4
If the chance to get a black bead is 1/5 the chance to get green or yellow has to be 4/5. Since both have the same amount of beads, the chance is 2/5 for each of them.
Now we know that 8 beads = 2/5 chance.
To find the number for 1/5 we just divide by 2.
8/2 = 4
Answer:
green Probability:2/5
yellow Probability:2/5
black Number of Beads:4
Step-by-step explanation best of luck
Which term could be put in the blank to create a fully simplified polynomial written in standard form?
Which term could be put in the blank to create a fully simplified polynomial written in standard form?
8x3y2−_____+3xy2−4y3
x2y2
x3y3
7xy2
7x0y3
The missing term in the polynomial is 8x³y² - __ + 3xy² - 4y³ is x²y².
What is a polynomial?A polynomial is an algebraic expression.
A polynomial of degree n in variable x can be written as,
a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² +...+ aₙ.
Given, A polynomial 8x³y² - __ + 3xy² - 4y³.
Now, If we observe the terms we can conclude 'x' has decreasing powers and y has increasing powers, therefore, the term could be put in the blank to create a fully simplified polynomial written in standard form is x²y².
So, The polynomial is 8x³y² - x²y² + 3xy² - 4y³.
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you come across a bomb that is about to go off if you do not stop it! to neutralize the bomb, you must get exactly four gallons of water in the empty tub to which it is attached. the only equipment you have is one empty 3-gallon jug, one empty 5-gallon jug and an unlimited supply of water. there are no markings on the jugs or the tub. the only thing you know is that one jug is 3-gallon and the other jug is 5-gallon. the jugs have to be completely filled. you cannot make eyeball assumptions about a half-filled jug. you have to make sure you get exactly 4 gallons in the tub. what do you do?
Fill the 3-gallon jug all the way.
Pour those 3 gallons of water into the 5-gallon jug.
Fill the 3-gallon jug all the way again, and pour as much as it takes to fill the 5-gallon jug (which is 2 gallons).
Now you have 5 gallons in the 5-gallon jug and 1 gallon in the 3-gallon jug.
Empty the 5-gallon jug.
Pour the 1 gallon in the 3-gallon jug into the 5-gallon jug.
Now fill the 3-gallon jug all the way, and pour all 3 gallons into the 5-gallon jug, which, added to the 1 gallon already in there, and leaves you with exactly 4 gallons.
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Fill the 3-gallon jug all the way.
Pour those 3 gallons of water into the 5-gallon jug.
Fill the 3-gallon jug all the way again, and pour as much as it takes to fill the 5-gallon jug (which is 2 gallons).
Now you have 5 gallons in the 5-gallon jug and 1 gallon in the 3-gallon jug.
Empty the 5-gallon jug.
Pour the 1 gallon in the 3-gallon jug into the 5-gallon jug.
Now fill the 3-gallon jug all the way, and pour all 3 gallons into the 5-gallon jug, which, added to the 1-gallon already in there, leaves you with exactly 4 gallons.
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Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. Estimate p(6) for n = 18 and p = 0. 3.
By applying normal approximation to the binomial concept, it can be concluded that the indicated probability is 0.62134
Normal approximation to the binomial is a method of approximating the probabilities associated with the binomial distribution using the normal distribution.
μ = n.p
σ = √np(1-p)
where μ is the mean, n is the number of trials, p is the success probability and σ is the standard deviation.
Now we have:
x = 6
n = 18
p = 0.3
Thus we can calculate the mean:
μ = np
= 18 * 0.3
= 5.4
Now we calculate the standard deviation
σ = √μ (1-p)
= √ 5.4 (1 - 0.3)
= √ 5.4 * 0.7
= √ 3.78
= 1.944
Then we can find the z-score using the mean and standard deviation found in the previous step:
z-score = (x – μ) / σ
= (6 - 5.4) / 1.944
= 0.6 / 1.944
= 0.309
And finally, we can obtain from the z table that P(6) = P(x<0.309) = 0.62134
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15 (x-3)/5= 3 (2x-3) without the distributive property ( the / symbol is division)
Answer:
brooooooooooo I got x=0
when a retired police officer passes away, he leaves $220000to be divided among his four children and four grandchildren. the will specifies that each child is to get twice as much as each grandchild. how much does each get?
Answer:
Each child gets $36666.67, each grandchild gets $18333.33
Step-by-step explanation:
220000/6= 36666.67,
36666.67/2= 18333.33
let f ( x ) = 2 x 3 and g ( x ) = 2x 2 4x . after simplifying, ( f ∘ g ) ( x )
for the functions f and g, fog is given by
fog (x) = 2 ( 2x ^2 +4x ) +3 = 4x^2 +8x+3
The equation of f(x) is f ( x ) = 2 x+ 3 and g ( x ) = 2x^ 2+ 4x
then fog is given by
fog (x) = 2 ( 2x ^2 +4x ) +3
= 4x^2 +8x+3
The graph of g(x) is shifted 3 units to the left. This is a horizontal translation.
For horizontal translation, we follow the following rule >>>
Let parent function be f(x), then
• f(x+a) is the parent translated "a" units left
• f(x-a) is the parent translated "a" units right for function
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Fill in the table using this function rule Y=3x+1
The complete table of values is
x 1 2 3 4
y 4 7 10 13
How to complete the tableFrom the question, we have the following parameters that can be used in our computation:
y = 3x + 1
Set the variable x from 1 to 4
So, we have
y = 3 * 1 + 1 = 4
y = 3 * 2 + 1 = 7
y = 3 * 3 + 1 = 10
y = 3 * 4 + 1 = 13
When represented on a table of values, we have
x 1 2 3 4
y 4 7 10 13
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You have a function that takes in an X value and produces a Y value. The x value equals 2 times the y value, plus 7 more.What's the x value when the y value equals 12? I need help
The x value when the y value equals 12 is calculated to be 31
How to find the value of xFrom the problem the equation is deduced to be : x = 2y + 7
When y = 12,
we can substitute this value into the equation to find the corresponding x value:
x = 2(12) + 7
x = 24 + 7
x = 31
Therefore, the x value when the y value equals 12 is 31.
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On the geography quizzes Elliot got 45 out of 54 countries correct and Lucy got an 88% and Jeanette got 38 out of 44 correct list the students standings from least to greatest
From least to greatest; Elliot, Jeanette, Lucy.
What is a Percentage?A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.
Elliot = 45/54 x 100%
Lucy = 88%
Jeanette = 38/44 x 100%
Elliot = 83%
Lucy = 88%
Jeanette = 86%
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Two students missed the first Mid exam. Their excuse was that their car had gotten a flat tire. On the next mid exam, these students were asked to identify which tire had become flat (driver's side front, driver's side rear, passenger's side front, passenger's side rear). What is the probability that the two students pick the same tire (assuming the students are randomly choosing)? Submit your answer as a decimal with four digits of accuracy.
Answer:
1/4=0.25
Step-by-step explanation:
The probability that both students pick the same tire is (1/4) * (1/4) = 1/16
The probability of the first student picking a specific tire is 1 in 4, or 1/4. The probability of the second student also picking that same tire is also 1 in 4. To find the probability that both students pick the same tire, we need to multiply the probabilities of each event.
So, the probability that both students pick the same tire is (1/4) * (1/4) = 1/16. This means that the chance of both students picking the same tire is 0.0625 or 6.25%.
To summarize, there are four possible tires that could have gotten a flat, and two students are asked to identify the tire. If each student randomly chooses a tire, the probability that both students will pick the same tire is 1 in 16, or 0.0625. This means that the chance of both students picking the same tire is low, and the likelihood of them choosing different tires is much higher.
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List the sample space for rolling a fair eight-sided die.
S = {1}
S = {8}
S = {1, 2, 3, 4, 5, 6}
S = {1, 2, 3, 4, 5, 6, 7, 8}
Answer: The sample space of rolling an eight-sided die would be (1, 2, 3, 4, 5, 6, 7, 8)
Step-by-step explanation: thank you
what are the roots of 3x^4-21x^2+18
Answer:
{-√6, -1, 1, √6}
Step-by-step explanation:
You want the roots of the quartic 3x^4-21x^2+18.
FactorsWe can find the roots by factoring to linear factors:
= 3(x^4 -7x^2 +6)
= 3(x^2 -6)(x^2 -1) . . . . . . . 6 = (-1)(-6) are factors with a sum of -7
= 3(x -√6)(x +√6)(x -1)(x +1) . . . . . a^2 -b^2 = (a -b)(a +b)
The roots of the quartic are ±√6 and ±1.
I’ll give Brainliest and points!!!!
* image is provided*
Plssss helpppp!!!!!!
Answer: C. 12000(1.05)^n < 20000
Step-by-step explanation:
When it says that the tuition is increasing at a rate of 5%, it means that you add 1
1+0.5= 1.05
Everything else is the same so you just add 1 to 0.5 and that is the answer.
The dog shelter has Labradors, Terriers, and Golden Retrievers available for adoption. If P(terriers) = 15%, interpret the likelihood of randomly selecting a terrier from the shelter.
Likely
Unlikely
Equally likely and unlikely
This value is not possible to represent probability of a chance event
The interpretation that the likelihood that the first question will be true is (a) Likely.
What is probability?Probability is a number that expresses the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
To interpret the likelihood that the first question will be true.
From the question, we have the following parameters that can be used in our computation:
Probability value
The value of the probability and the notation are given as
P(true) = 0.8
As a general rule of probability, we have
Range of probability value = 0 to 1 (inclusive)
Unlikely = less than 0.5
Equally likely and unlikely = 0.5
Likely = greater than 0.5
Using the above as a guide, we have the following:
The value of P(true) = 0.8 is greater than 0.5
This that the probability is likely
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find an equation of the sphere with center and radius . what is the intersection of this sphere with the -plane?
The intersection of this sphere with the -plane is (y+6)²+(z-4)²=21
The equation of a sphere centered at (h,k,l) and radius r is given by:
(x−h)²+(y−k)²+(z−l)²=r²
where x,y, and z represent the coordinates of the points on the surface of the surface.
so if the center is (-3,2,5) and radius r=4 then we can plug the values in the above formula:
consider the points h=-3 k=2 l=5 and r=4.
(x−(-3))²+(y−2)²+(z−5)²=4²
(x+3)²+(y−2)²+(z−5)²=16.
If the sphere intersects the yz-plane then x=0
The resulting equation is an equation of a circle with a radius r=√21
and at the center with (6,-4) on the yz-plane
with yx-plane:
⇒(x−2)²+(y+6)²+(z−4)²=25
⇒ (0−2)²+(y+6)²+(z−4)²=25
⇒4+(y+6)²+(z−4)²=25
⇒(y+6)²+(z−4)²=25-4
⇒(y+6)²+(z−4)²=21
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Write the log equation
as an
exponential equation. You do not need to solve for x.
In (4) = 3x +1
Answer:
Step-by-step explanation:
Ln means log to the base e and 3x+1 is an exponent
So the answer is:
4 = e^(3x+1).
Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number (10)10. (11)11. (13)13 is equal to_____
The number of divisors of 1730 of the form 4n + 1 is 20.
What do you mean by integers?An integer is a whole number, meaning it can be positive, negative, or zero, but it cannot be a fraction or a decimal. Integers are a subset of real numbers, which also include fractional and decimal numbers.
Integers are commonly used in mathematics and computer science for counting and for performing arithmetic operations such as addition, subtraction, multiplication, and division. They are also used to represent the magnitude of a quantity, such as the number of items in a set or the time elapsed in seconds.
In computer programming, integers are often stored as binary numbers, which are sequences of bits that represent positive or negative whole numbers. Different computer architectures have different numbers of bits for representing integers, with common sizes being 8, 16, 32, and 64 bits.
The number (10)10. (11)11. (13)13 is equal to 11 × 13 × 10 = 1730 in decimal.
A divisor of a number can be represented as (4n + 1) where n is a non-negative integer.
To find the number of divisors of 1730 of the form 4n + 1, we need to find the number of factors of 1730 that can be expressed as 4n + 1.
First, we find the prime factorization of 1730:
1730 = 2 × 3 × 5 × 29
Since 1730 is an odd number, one of its divisors must also be an odd number.
The only possible values for n are 0, 1, 2, ..., (number of odd divisors - 1).
In this case, the number of odd divisors of 1730 can be calculated as:
(number of ways to choose 0 odd factors) × (number of ways to choose 1 odd factor) × (number of ways to choose 2 odd factors) × ... × (number of ways to choose 4 odd factors)
= 1 × 4 × 5 × 1 = 20
Thus, the number of divisors of 1730 of the form 4n + 1 is 20.
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What is the lenght of MN? Important to have calculator in degree mode
Round your answer to the tenths
The length of side MN from triangle MNP is 30.78
Please see the attached picture below as reference of triangle MNP.
From the case, we know that:
∠M = 90°
∠P = 72°
∠N = 18°
PM = 10
To solve this problem we need to find the length of side NP first using cos formula to angle P.
Cos ∠P = PM
NP
Cos 72° = 10 / NP
0.309 = 10 / NP
NP = 32.36
Next, we will use the same approac to angle N:
Cos ∠N = MN
NP
Cos 18° = MN / 32.36
MN = 0.951 x 32.36
MN = 30.78
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Raj polled 100 students in his class to compare the number of hours that teen boys and girls played video games each week. The results of his survey are shown below. Which statement about his data sets is true?
The statement about Raj's data sets which true is The average boy plays 8.5 hours more per week than the average girl.
About data setTechnically, a dataset is a collection of related items that can be accessed individually or in certain management combinations as a single unit. Datasets can be organized into several types of data structures. Examples of dataset in the business world can be seen from names, salaries, employee contact information, to sales figures, and so on.
In conclusion, a dataset is a collection of data that is ordered and obtained from a collection of information. The collection of information itself is obtained from observation, measurement, study, or analysis to become data. Data can be facts, numbers, names, or even descriptions. Therefore, datasets are closely related to data mining activities which help data scientists analyze data into coherent information.
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which of the following statements is least accurate? multiple choice question. the height of each rectangle represents cumulative frequency or cumulative relative frequency. the rectangles of a histogram represent grouped data. the rectangles of a histogram are drawn with no space, or gaps, between them. the rectangles of a histogram represent both the class width and frequency, or relative frequency, of the respective class.
The following statements is least accurate: When adding and subtracting, consider the number of buildings instead of Sf.
What does less accurate mean?
When adding or subtracting numbers, the number of digits in the result must be equal to the smallest number of digits after the decimal point of a term in the sum. How closely a measured value agrees with a reference or known value is referred to as accuracy. For example, if you get a weight measurement of 3.2 kg for a substance in a laboratory, but the actual or known weight is 10 kg, your measurement is not accurate.In this case, your measurement is nowhere near the known value.
Typically, bottom-up estimates fall into this category: the most accurate estimate is the final estimate because it is created by estimating individual costs for different parts of the project and then combining them into one estimate.
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write the fully simplified negation of 3 < x ≤ 4
The given inequality states that the value of x is greater than 3 and less than or equal to 4.
3 ≥ x > 4
Given inequality: 3 < x ≤ 4
Negating the given inequality:
1. Negate the inequality symbol:
3 ≥ x
2. Negate the comparison operator:
3 ≥ x > 4
The given inequality states that the value of x is greater than 3 and less than or equal to 4. When we negate this inequality, we get that the value of x is greater than or equal to 3 and greater than 4. This means that the fully simplified negation of 3 < x ≤ 4 is 3 ≥ x > 4.
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Eric throws a biased coin 10 times and gets tails 3 times.
Sue throws the same coin 50 times and gets tails 20 times.
Audi is going to throw the coin once.
Use Erics and Sues results to work out an estimate for the probability that Aadi will get tails. write your fraction in the form a/b.
Answer:
Below
Step-by-step explanation:
Out of 50 + 10 = 60 tries, 3 + 20 = 23 times were tails
23/60 chance of tails
The estimate of the probability that Audi gets tails is given as follows:
p = 23/60.
How to calculate a probability?A probability is calculated via the division of the number of desired outcomes by the number of total outcomes.
Hence the probability of each person getting tails when they throw a biased coin is given as follows:
Eric: 3/10.Sue: 20/50 = 2/5.The total number of throws is given as follows:
50 + 10 = 60.
The number of tails is given as follows:
3 + 20 = 23.
Hence the probability for Audi is given as follows:
p = 23/60.
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At what point on the curve x = 3t^2 + 5, y = t^3 − 7 does the tangent line have slope 1/2 ?
The tangent line on the curve x = 3t² + 5 , y = t³ - 7 have a slope as 1/2 at the point (8, -6).
The Tangent Line to a curve at a specific point is calculated by the derivative of the curve at that point.
The Slope of the tangent line at a point (x₀, y₀) is given by the derivative of the curve at that point, denoted as f'(x₀).
The curve x = 3t² + 5 , y = t³ - 7 ,
we have to find the value of t such that the slope of the tangent line at that point is 1/2.
we first find the derivative of y with respect to x, which is given by:
⇒ dy/dt = 3t² and dx/dt = 6t
Using chain rule to find the derivative of y with respect to x:
⇒ dy/dx = dy/dt × dt/dx = (3t²)/6t = t/2
Next, we equate derivative to 1/2 and solve for t ;
⇒ t/2 = 1/2
⇒ t = 1
So, the point where the tangent line has slope 1/2 is
⇒ (x, y) = (3t² + 5, t³ - 7)
⇒ (8, -6).
Therefore , the point is (8,-6) .
The given question is incomplete , the complete question is
At what point on the curve x = 3t² + 5 , y = t³ - 7 , does the tangent line have slope as "1/2" ?
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