To find the rate of change of the x-coordinate at the point (1, 5) with a given y-coordinate rate of change, we use the chain rule to differentiate the equation y = 24 + [tex]x^3[/tex] with respect to time, and solve for dx/dt.
Explanation:The student's question involves the relationship between the rates of change of the x-coordinate and y-coordinate of a particle moving along a curve. Given the equation y = 24 +[tex]x^3[/tex], we need to find how fast the x-coordinate is changing at the point (1, 5), given that the y-coordinate is increasing at a rate of 4 cm/s. To find this, we can use the chain rule from calculus to relate the rates of change. Differentiating both sides with respect to time t, we get dy/dt = 3[tex]x^2[/tex] dx/dt. Solving for dx/dt, we find the rate of change of the x-coordinate when x = 1 and dy/dt = 4 cm/s.
At the point (1, 5), [tex]\( \frac{{dx}}{{dt}} = \frac{4}{3} \)[/tex] cm/s when [tex]\( \frac{{dy}}{{dt}} = 4 \)[/tex] cm/s.
To find how fast the x-coordinate is changing [tex](\( \frac{{dx}}{{dt}} \))[/tex] when the y-coordinate is increasing [tex](\( \frac{{dy}}{{dt}} \)),[/tex] we'll use implicit differentiation.
Given [tex]\( y = 24 + x^3 \),[/tex] differentiate both sides with respect to time [tex](\( t \)):[/tex]
[tex]\[ \frac{{dy}}{{dt}} = \frac{{d}}{{dt}}(24 + x^3) \][/tex]
Given that [tex]\( \frac{{dy}}{{dt}} = 4 \)[/tex], and when [tex]\( x = 1 \) and \( y = 5 \),[/tex] we can find [tex]\( \frac{{dx}}{{dt}} \):[/tex]
[tex]\[ 4 = 0 + 3x^2 \cdot \frac{{dx}}{{dt}} \][/tex]
[tex]At \( x = 1 \):[/tex]
[tex]\[ 4 = 3(1)^2 \cdot \frac{{dx}}{{dt}} \][/tex]
[tex]\[ 4 = 3 \cdot \frac{{dx}}{{dt}} \][/tex]
[tex]\[ \frac{{dx}}{{dt}} = \frac{4}{3} \][/tex]
So, the x-coordinate of the point is changing at a rate of [tex]\( \frac{4}{3} \)[/tex] cm/s at that instant.
Can someone please be kind enough to help me with all theses problems
how can you figure out how many miles you are using on a united fligjts
which of the following is a linear equation: (a)x2-y2=1 (b)y=3/4-x/2 (c)x2+y2=1 (d)y=3/4+2/x
Solve for x. Show each step of the solution. 4.5(8−)+36=102−2.5(3+24) i need to show my work dont just give me the answer
Find the measure of an angle if its measure is 46 degrees less than three times its complement.
How many real solutions does the equation 0.2x^5-2x^3+1.8x+k=0 have when k = 0?
Answer:
It is 5
When Y=1 I got 0
Is there a value of k for which the equation has just one real solution?
I put no
Then for the last one I put Yes
Step-by-step explanation:
Edg 2020
compare discrete domains and continuous domains.
Rename 12,000 ten thousand
Renaming 12,000 as 'ten thousand' means expressing it as '1 ten thousand and 2 thousand'. This breaks the number down into units of ten thousand, with 10,000 being the base unit and the additional 2,000 making up the rest.
Explanation:The phrase 'rename 12,000 ten thousand' refers to the process of expressing the number 12,000 in terms of ten thousand. To do this, we need to understand that the number 10,000 is a base unit in this context, acting as one 'ten thousand'. Therefore, the number 12,000 would be considered as 1 ten thousand and 2 thousand, since 12,000 is made up of one 10,000 and an additional 2,000 units.
To better understand, let's do a breakdown:
10,000 (one ten thousand) + 2,000 (two thousand) = 12,000We know that:
10,000 is one ten thousand.20,000 is two ten thousands.30,000 is three ten thousands.And so on, up to 160,000, which would be sixteen ten thousands.In this case, when we want to 'rename' 12,000 as ten thousand, it becomes 1 ten thousand and 2 thousand.
PLEASE HELP! I'm so confused!
Hourly employees working more than 40 hr a week are often paid "time and a half" for overtime, any hours beyond 40. In the formula p = wr + 1.5we, gross pay (p) is computed by multiplying the hourly wage (w) by the number of regular hours worked (r[tex] \leq [/tex]40) and adding 1.5 times the product of the hourly wage (w) and the number of extra hours (e) beyond 40. Find your gross pay if you work the given number of hours at the given rate.
40 hr at $8.12/hr
47 hr at $10.92/hr
1. Is the equation true, false, or open? 9p + 8 = 10p + 7 A. Open; there is a variable. B. True; the expressions are the same for all values of the variables. C. False; the expression are never the same.
Also all the answers in the Unit Test please
For spring break you and some friends plan a road trip to a sunny destination that is 1705 miles away. if you drive a car that gets 39 miles per gallon and gas costs $3.339/gal, about how much will it cost to get to your destination?
Answer:
$145.98
Step-by-step explanation:
You and some friends plan a road trip to a sunny destination that is 1705 miles away.
You drive a car that gets 39 miles per gallon.
Total gas used in the trip = [tex]\frac{1705}{39}[/tex]
= 43.7179487 ≈ 43.72 gallons
Gas costs per gallon = $3.339
Total cost to get to your destination = 43.72 × 3.339
= $145.98108 ≈ $145.98
The cost would be $145.98 to get to your destination.
Y2 + 30 ÷ x - 8 what is the value of the expression when x = 6 and y = 8?
Can someone help please? Thank you. SHOW ALL WORK.
Please help! 20 points!
divide: 5/4 = 1.25
answer is 1.25
Please help me solve this equation but show the steps so I can learn how to solve it.
0.25r-0.125+0.5r=0.5+r
What are the zeros of the polynomial function y = x(x − 6)(x + 3)(x − 11)?
Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.
7x – 4y = 4
x – 4y = 3
A. Perpendicular
B. Parallel
C. Neither
A rectangle patio is surrounded on three sides by a fence. If the areas of the patio is 45 m sq and the total of fence is 19m what is the length and width of the patio?
PLZ HELP Simplify the rational expression. State any excluded values. 2x/-5x+x^2
Answer:
The excluded value is x=5.
Step-by-step explanation:
Given : Expression [tex]\frac{2x}{-5x+x^2}[/tex]
To find : Simplify the rational expression. State any excluded values.
Solution :
First we simplify the expression,
[tex]\frac{2x}{-5x+x^2}[/tex]
Taking x common from the denominator,
[tex]=\frac{2x}{x(-5+x)}[/tex]
Cancel the common term from numerator and denominator,
[tex]=\frac{2}{-5+x}[/tex]
To find the excluded value we put the denominator equal to zero to get the value of x in which given expression is not valid.
[tex]-5+x=0[/tex]
[tex]x=5[/tex]
Therefore, The excluded value is x=5.
Please help me thank you very much.
Please answer all 4 questions.
Which type of question fits a binomial model? A. fill in the blank B. matching C. multiple choice D. true/false
The type of question that fits a binomial model is a true/false question, as this type of question strictly limits the outcome to two possibilities - correct or incorrect.
Explanation:The type of question that fits into a binomial model would be a true/false question. A binomial distribution, in mathematics and statistics, is used to model a number of successes in a sequence of independent and identically distributed binary experiments, such as flipping a coin or indeed, answering a true/false question. These types of questions are considered binomial because there are two possible outcomes: right or wrong, true or false. On the other hand, options like 'fill in the blank', 'matching', or 'multiple choice' doesn't fit the binomial model because they do not strictly limit the outcome to two possibilities.
Learn more about Binomial Model here:https://brainly.com/question/33930713
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What is y=-4x-15 in standard form
Final answer:
To convert the equation y = -4x - 15 to standard form, you add 4x to both sides to get 4x + y = -15, which aligns with the standard form Ax + By = C.
Explanation:
To convert the equation y = -4x - 15 into standard form, we need to rearrange it such that the x- and y-terms are on one side of the equation and the constant is on the other side, typically with the x term first. The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A should be non-negative.
The given equation is already close to this form, but we need to get the x-term to the other side of the equation. To do this, we simply add 4x to both sides, giving us:
4x + y = -15
This is the standard form of the given linear equation.
Factor out the greatest common factor. If the greatest common factor is 1, just retype the polynomial.
4t^3+8t
The school store sells packs of 12 pens for $2.40. Which is NOT a unit rate to describe this sale?
A. 20 cents per pen
B. 5 pens per dollar
C. 2.40 dollars per pack of pens
D. 120 pens per 24 dollars
Solve for x: 3x − 5 = 2x + 6.
1
−1
11
−11
Answer:
X = 11 sweetheart
A farmer drives a tractor from one town to another, a distance of 120 kilometers. He drives 10 kilometers per hour fast on the return trip, cutting an hour off the time. How fast does he drive each way?
Determine the nature of the roots of the function
f(x)=7x^2+5x-1
Final answer:
To determine the nature of the roots of the function f(x)=7x²+5x-1, we calculate the discriminant, which is 53. Since it's positive, the equation has two distinct real roots.
Explanation:
The nature of the roots of the quadratic function f(x)=7x²+5x-1 can be determined by calculating the discriminant, which is part of the quadratic formula. The discriminant is given by b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. For the given function, a = 7, b = 5, and c = -1. Therefore, the discriminant equals:
D = b² - 4ac
D = (5)² - 4(7)(-1)
D = 25 + 28
D = 53
Since the discriminant is positive (D > 0), the quadratic function has two distinct real roots.
There are 84 boys in a senior class of 146 students. find the ratio of boys to girls. (note: if you know the total number of student in the class and the number of boys, what do you need to do to find the number of girls?
The graph shown is the solution set for which of the following inequalities? y ≥ x + 1 y ≥ |x| + 1 y ≥ |x - 1|
Answer:
y ≥ |x| + 1
Step-by-step explanation:
I did this on odyssey ware and got 100% so YES it is correct!
What is the tangent of angle a?
a. 6/8
b. 8/10
c. 8/6
d. 6/10