x=-3
Step-by-step explanation:Solve:9x+2(3x+7)= -31
multiply the "3x" and the "7" by two (distributive property)
[="6x" and "7"]
add the "9x" and the "6x" [="15x"]
subtract "14" from"-31" [= "-45"]
divide "15x" from "-45"
="-3"
Check:put the "-3" in for "x"
multiply 9(-3) [=-27]
multiply 3(-3) [=-9]
distribute the 2 to the "-9" and the "7" [="-18" and "14"]
add the "-18" and "14" [=-4]
add the "-27" and the "-4"
=-31 ✓
What it looks like:Solve:9x+2(3x)+2(7)= -31
9x+6x+14= -31
15x+14= -31
-14 -14
15x= -45
/15= /15
x= -3
Check:9x+2(3x+7)= -31
9(-3)+2(3(-3)+7= -31
-27+2(-9+7)= -31
-27+(2(-9)+2(7))= -31
-27+(-18+14)= -31
-27+ (-4)= -31
-31= -31✓
A publishing industry report predicts that magazine subscriptions will drop
by 10% each year. If this prediction is correct, then how many subscribers
will a magazine with 51,000 subscribers have in 7 years?
If necessary, round your answer to the nearest whole number.
Answer:
24,393 subscribers
Step-by-step explanation:
This is exponential decay problem. The formula for exponential decay is:
[tex]F=P(1-r)^t[/tex]
Where
F is the future value (what we are looking for, in 7 years)
P is the present amount (that is 51,000)
r is the rate of decrease per year (r = 10% = 10/100 = 0.1)
t is the time, in years (t = 7)
Now substituting, we get:
[tex]F=P(1-r)^t\\F=51,000(1-0.1)^7\\F=51,000(0.9)^7\\F=24,393.14[/tex]
Rounding to nearest whole number,
Number of subscribers after 7 years = 24,393 subscribers
Using the exponential decay formula, a magazine with an initial 51,000 subscribers will have approximately 24,393 subscribers left in 7 years if subscriptions decline by 10% annually.
Explanation:The question is asking to predict the number of magazine subscribers after a period of 7 years, assuming a 10% annual decline. To calculate this, we can use the formula for exponential decay, which is A = P(1 - r)^t, where A is the amount after time t, P is the initial principal amount (51,000 subscribers), r is the rate of decline (10% or 0.10), and t is the time in years (7 years). Calculating this, we get:
Initial subscribers: 51,000Annual decline rate: 10%Number of years: 7Subscribers after 7 years: A = 51,000(1 - 0.10)^7Now, calculate the value of A:
A = 51,000(1 - 0.10)^7
A = 51,000(0.90)^7
A = 51,000(0.4782969)
A ≈ 24,393 subscribers (after rounding to the nearest whole number).
Therefore, if the magazine subscriptions continue to decline at a rate of 10% per year, the magazine is predicted to have approximately 24,393 subscribers in 7 years.
A cube is a special right rectangular prism with congruent sides. If a block of cheese is shaped like a cube with a length of 5 inches, find the volume of the block of cheese.
Answer:
125 cubic inches
Step-by-step explanation:
The formula for the volume of a cube is:
[tex]V=a^3[/tex],
Where V is volume and a is edge length. Let's substitute the edge length of 5 from the problem into this formula.
[tex]V=5^3[/tex]
Now we can simplify. I will rewrite 5 cubed as 5*5*5 for simplicity.
[tex]V=5*5*5[/tex]
[tex]V=125[/tex]
So we have our volume, 125, but what does this mean? We have to figure out the units. Recall that our original unit was inches, thus the final answer is in inches. However, 125 inches does not make much sense, as we are dealing with volume, and not a line or distance. 125 cubic inches is the answer we are looking for, as cubic inches is the unit for volume deriving from inches.
The volume of a block of cheese is 125 in³.
What is a cube?It is a polygon having six faces.
The volume of a cube is side³.
We have,
Side length = 5 in
Now,
The volume of a block of cheese.
= Side³
= 5³
= 125 in³.
Thus,
125 in³ is the volume of a block of cheese.
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What is the simplified expression for the expression below?
-1(2x + 3) – 2(x - 1)
4x + 1
4x -2
ooo
4x+2
4x
-
1
Answer:
-4x - 1Step-by-step explanation:
[tex]-1(2x+3)-2(x-1)\qquad\text{use the distributive property}\\\\=(-1)(2x)+(-1)(3)+(-2)(x)+(-2)(-1)\\\\=-2x-3-2x+2\qquad\text{combine like terms}\\\\=(-2x-2x)+(-3+2)\\\\=-4x-1[/tex]
Answer:
-4x-1
Step-by-step explanation:
-1(2x+3)-2(x-1)
-2x-3-2x+2
-2x-2x-3+2
-4x-3+2
-4x-1
When adding negative and positive numbers, you can use the associative and commutative properties to group all of the positive addends and then group all the negative addends. Which expression shows the second step of this method?
5 + (-1) + (-5) + (-2) + 11 + (-7) + 2
Answer:
The second step of this method is
[tex]5+(-1)+(-5)+(-2)+11+(-7)+2=(5+11+2)-(1+5+2+7)[/tex]
The value of the given expression is
[tex]5+(-1)+(-5)+(-2)+11+(-7)+2=3[/tex]
Step-by-step explanation:
Given expression is [tex]5+(-1)+(-5)+(-2)+11+(-7)+2[/tex]
To find the value of the given expression [tex]5+(-1)+(-5)+(-2)+11+(-7)+2[/tex]:
Arranging positive and negative addends by using the associative and commutative properties of the given expression as below:
[tex]5+(-1)+(-5)+(-2)+11+(-7)+2=5+11+2+(-1)+(-5)+(-2)+(-7)[/tex]
[tex]=(5+11+2)-(1+5+2+7)[/tex]
[tex]=18-15[/tex]
[tex]=3[/tex]
[tex]5+(-1)+(-5)+(-2)+11+(-7)+2=3[/tex]
The second step of this method is
[tex]5+(-1)+(-5)+(-2)+11+(-7)+2=(5+11+2)-(1+5+2+7)[/tex]
The value of the given expression is
[tex]5+(-1)+(-5)+(-2)+11+(-7)+2=3[/tex]
Solve for x:
2x-4y=12
Answer:
x = 2y + 6
Step-by-step explanation:
2x - 4y = 12
2x = 4y + 12
x = 2y + 6
To solve for 'X' in the equation 2x - 4y = 12, isolate 'X' by adding 4y to both sides, resulting in the equation 2x = 4y + 12. Further, divide the entire equation by 2 to get X = 2y + 6. The final value of 'X' would depend on the value of 'Y'.
To solve for X in the equation 2x - 4y = 12, we can proceed in the following manner:
Isolate X on one side of equation. In order to do this, we add 4y to both sides of the equation, thus changing the equation to 2x = 4y + 12.
Now, to solve for X, we need to divide the entire equation by 2. Therefore, the final equation is ready: X = 2y + 6.
Thereby, you have now solved for 'X' in the equation. Kindly note that as there is a 'y' variable involved as well, the final value of 'X' would depend on the value of 'Y'.
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?? “solve the missing elements for each problem. use 3.14 for area
Answer:
Problem 4:
Radius = 11 cm.
Circumference = 69.08 cm.
Problem 5:
Radius = 14 mm.
Circumference = 87.92 mm.
Problem 6:
Radius = 9 inches.
Circumference = 56.52 inches.
Problem 7:
Diameter = 32 ft.
Circumference = 100.48 ft.
Problem 8:
Diameter = 26 yards.
Circumference = 81.64 yards.
Problem 9:
Diameter = 12 cm.
Circumference = 37.68 cm.
Step-by-step explanation:
Problem 4:
Given:
Diameter GH = 22 cm
We need to find the radius EF and Circumference of the circle.
Now radius is given by half of the diameter
Also Circumference is given by π times diameter.
Radius EF = [tex]\frac{1}{2}\times Diameter = \frac{1}{2}\times 22 = 11 \ cm[/tex]
Circumference = [tex]\pi \times Diameter = 3.14 \times 22 = 69.08\ cm[/tex]
Hence Radius = 11 cm
Circumference = 69.08 cm.
Problem 5:
Given:
Diameter OP = 28 mm
We need to find the radius MN and Circumference of the circle.
Now radius is given by half of the diameter
Also Circumference is given by π times diameter.
Radius MN = [tex]\frac{1}{2}\times Diameter = \frac{1}{2}\times 28 = 14 \ mm[/tex]
Circumference = [tex]\pi \times Diameter = 3.14 \times 28 = 87.92\ mm[/tex]
Hence Radius = 14 mm
Circumference = 87.92 mm.
Problem 6:
Given:
Diameter EF = 18 inches
We need to find the radius CD and Circumference of the circle.
Now radius is given by half of the diameter
Also Circumference is given by π times diameter.
Radius CD = [tex]\frac{1}{2}\times Diameter = \frac{1}{2}\times 18 = 9 \ in[/tex]
Circumference = [tex]\pi \times Diameter = 3.14 \times 18 = 56.52\ in[/tex]
Hence Radius = 9 inches.
Circumference = 56.52 inches.
Problem 7:
Given:
Radius GH = 16 ft
We need to find the Diameter CD and Circumference of the circle.
Now Diameter is given by twice the radius.
Also Circumference is given by π times diameter.
Diameter KJ = [tex]2 \times radius = 2 \times 16 = 32 \ ft[/tex]
Circumference = [tex]\pi \times Diameter = 3.14 \times 32 = 100.48\ ft[/tex]
Hence Diameter = 32 ft.
Circumference = 100.48 ft.
Problem 8:
Given:
Radius LM = 13 yards
We need to find the Diameter CD and Circumference of the circle.
Now Diameter is given by twice the radius.
Also Circumference is given by π times diameter.
Diameter NO = [tex]2 \times radius = 2 \times 13 = 26 \ yards[/tex]
Circumference = [tex]\pi \times Diameter = 3.14 \times 26 = 81.64\ yards[/tex]
Hence Diameter = 26 yards.
Circumference = 81.64 yards.
Problem 9:
Given:
Radius NO = 6 cm
We need to find the Diameter CD and Circumference of the circle.
Now Diameter is given by twice the radius.
Also Circumference is given by π times diameter.
Diameter PQ = [tex]2 \times radius = 2 \times 6 = 12 \ cm[/tex]
Circumference = [tex]\pi \times Diameter = 3.14 \times 12 = 37.68\ cm[/tex]
Hence Diameter = 12 cm.
Circumference = 37.68 cm.
Which graph show the solution set for -5/2x -3<2
The graph that shows the solution set for the inequality -5/2x -3<2 is a number line with an open circle at -2 and an arrow pointing to the left.
What is inequality?Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
To graph the solution set for the inequality -5/2x -3<2, you first need to solve the inequality for x.
-5x -6 < 4
-5x < 4 + 6
-5x < 10
x < -2
The graph of the solution set for the inequality x < -2 is a number line with an open circle at -2 and an arrow pointing to the left. The open circle indicates that -2 is not included in the solution set, and the arrow pointing to the left indicates that all values to the right of -2 are included in the solution set.
Hence, the correct answer would be an option (C).
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Mel buys paper plates and paper cups for an event.
The plates are sold in packs. Each pack contains 60 plates.
The cups are sold in packs. Each pack contains 36 cups.
Mel buys exactly the same number of plates as cups.
What is the least number of each pack that Mel buys?
Answer:
3 packs of plates; 5 packs of cups
Step-by-step explanation:
First we have to find the least common multiple of the two. (factor tree, birthday cake) (Strategies not shown) You should get 180.
Next we have to see how many times each number goes into 180.
60 times 3 is 180 plates. 36 times 5 is 180.
That's your answer. Hope this helps!
he sum of three consecutive numbers is 72. What is the largest of these numbers?
Answer: The three consecutive numbers are 23, 24 and 25, and 25 Is obviously the largest.
Step-by-step explanation:
Answer: 23 , 24 , 25
Step-by-step explanation:
Consecutive numbers are numbers that follow each other in order.
Let the first number be x , the second number be x + 1 and the third number number be x + 2.
From the first statement :
Their sum implies :
x + x +1 + x + 2 = 72
3x + 3 = 72
Subtract 3 from both sides of the equation:
3x + 3 - 3 = 72 - 3
3x = 69
Divide through by 3
3x/3 = 69/3
Therefore :
x = 23
Therefore :
The numbers are
23 , 24 and 25
4. Simplify (2 – 3x)(3x2 - 2x -1).
A gx3 - X-2
B-9x² + 12x² - - 2
C 6x² + 4x + 3
D 6x3 - 13x2 + 4x + 3
Answer:
--9x^3+12x^2-x-2
Step-by-step explanation:
(2-3x)(3x^2-2x-1)
6x^2-4x-2-9x^3+6x^2+3x
-9x^3+6x^2+6x^2+3x-4x-2
-9x^3+12x^2-x-2
Mae Ling earns a weekly salary of $315 plus a 5.5% commission on sales at a gift shop. How much would she make in a work week if she sold $4,700 worth of merchandise?
She would make $573.5 in a work if she sold $4,700 worth of merchandise
Step-by-step explanation:
The given is:
Mae Ling earns a weekly salary of $315 plus a 5.5% commission on sales at a gift shopShe sold by $4,700We need to find how much she would make in a work week
∵ Mae Ling earns a weekly salary of $315
∵ She earns a 5.5% commission on sales
∵ She sold by $4,700 in that week
- Add $315 to the product of 5.5% and $4,700
∴ She made = 315 + (5.5% × 4,700)
∵ 5.5% = 5.5 ÷ 100 = 0.055
∴ She made = 315 + (0.055 × 4,700)
∴ She made = 315 + 258.5
∴ She made = 573.5
She would make $573.5 in a work if she sold $4,700 worth of merchandise
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Cindy worked 4 hours each day for 12 days. She earned a total of $240. How much did Cindy earn per hour?
Answer:
$5 per hour
Step-by-step explanation:
12*4=48 hours
240/48=5
4 times the sum of a number and -3 is 4 more than twice the number. Write and solve an equation to find the number.
Answer:
The number is 8.
Step-by-step explanation:
4(x+(-3))-4=2x
4(x-3)-4=2x
4x-12-4=2x
4x-2x-16=0
2x-16=0
2x=0+16
2x=16
x=16/2
x=8
You plant a spruce tree that is
is 8 inches tall that grows 6 inches a year.
ches a year and a hemlock tree that
a.write a system of linear equations that represent this situation.
b. Solve the system using any method.
Answer:
Y = 8 + 6X is correct answer.
Step-by-step explanation:
Let y be the total lenght of tree and x is number of years that has past after tree plantation.
So the total lenght of tree after x years can be presented with the help of equation given below.
Y = 8 + 6X
* This equation show that there is direct relationshop between number of years and total height of tree.
For example after 3 years the height of three will be
Y = 8 + 6(3) = 26 inches
Matt has started a car washing business to save money for college. Each week, he
washes 3 more cars than he did the week before.
Answer:
It’s incomplete
Step-by-step explanation:
Answer:
Part A: An = 6 + 3( n - 1 )
Part B: 36
explanation:
We us the explicit rule 6 + 3( n - 1 ) to find out the he found in the 11th week
6 + 3( n - 1 )
6 + 3( 11 - 1 )
6 + 3( 10 )
6 + 30
36
Hope it helps!
Four people share 2/3 pounds of peanuts equally.What fraction of a pound of peanuts does each person receive
Answer: 1/6 of a pound.
Step-by-step explanation: Divide 2/3 pounds by 4 people. To do this, multiply 2/3 by 4's reciprocal, which is 1/4. The reciprocal of a number is basically the reverse form of that number. 4 as a fraction is 4/1. The reciprocal of that would be 1/4. Hopefully you understand.
2/3 times 1/4 is 2/12. Simplify by dividing 2/12 by 2 and 12's greatest common factor: 2. 2 divided by 2 is 1; 12 divided by 2 is 6. Your final answer is 1/6.
Select the correct answer.
How do you express a gain of 4 yards in a football game as a rational number?
OA. 4.5 yards
OB. +4 yards
OC. I yards
OD. - yards
E. - 4 yards
Answer:
B
Step-by-step explanation:
if you gain yards its plus 4 yards or how ever many yard to gain or lose.
Answer:
+4
Step-by-step explanation:
Because you GAIN 4
A case of twenty-four 7.8 ounce containers of Greek yogurt costs $12.99.
Calculate the unit price of the Greek yogurt (cost per ounce). Round to the
nearest cent.
Answer:
Cost per ounce = $0.07 (7 cents per ounce)
Step-by-step explanation:
Each container is 7.8 ounces and there are 24 of them in a case.
Total weight in ounce of the case is:
7.8 * 24 = 187.2 ounces
It costs $12.99 for the case, which contains 187.2 ounces. We find the cost per ounce of yogurt by dividing the total cost by total ounces in the case.
12.99/187.2 = 0.0693
Rounded to nearest cent (2 decimal places), that would be:
Cost per ounce = $0.07 (7 cents per ounce)
The unit price of the Greek yogurt is $0.07 per ounce.
To calculate the unit price, we need to divide the total cost of the case by the total number of ounces in the case.
First, we determine the total number of ounces in the case by multiplying the number of containers by the ounces per container:
[tex]\[ 24 \text{ containers} \times 7.8 \text{ ounces per container} = 187.2 \text{ ounces} \][/tex]
Next, we divide the total cost of the case by the total number of ounces to find the cost per ounce:
[tex]\[ \frac{\$12.99}{187.2 \text{ ounces}} \approx \$0.0693 \text{ per ounce} \][/tex]
Finally, we round this to the nearest cent, which gives us:
[tex]\[ \$0.0693 \text{ per ounce} \approx \$0.07 \text{ per ounce} \][/tex]
Therefore, the unit price of the Greek yogurt, rounded to the nearest cent, is $0.07 per ounce.
Given the functions below, find (f.g) (-1)
f(x) = x² + 3
g(x) = 4x - 3
To calculate the composite function (f ∙ g)(-1) for the functions f(x) = x² + 3 and g(x) = 4x - 3, we first find g(-1) which is -7, and then calculate f(-7) which results in 52.
Explanation:Composite FunctionsTo find the composite function (f ∙ g)(x), we first apply the function g to x, and then apply the function f to the result of g(x). In this case, first find g(-1) and then apply f to that result.
Initialize with g(-1), which is 4(-1) - 3. Simplifying that, we get -4 - 3, which equals -7.
Now, take the result of g(-1), which is -7, and input it into f(x) to get f(g(-1)). Thus, f(-7) = (-7)² + 3. Calculating this gives us 49 + 3, which equals 52. Therefore, (f ∙ g)(-1) = 52.
Write (3+4 i)+(8+2 i)as a complex number in standard form
Answer:
11+6i
Step-by-step explanation:
(3+4i)+(8+2i)=11+6i
Explain to me what x is in this problem: 2/3=x/32
Answer:
x=21.33....
Step-by-step explanation:
2/3=x/32
cross product
3*x=2*32
3x=64
x=64/3
x=21.3
Last year, Maria spent $2,388 for natural gas. This year, she decided to use balance billing. What will her monthly payment be this year?
Answer:
[tex]\$ 199[/tex]
Step-by-step explanation:
Given: Maria spent $2388 for natural gas, last year.
Maria chooses to use balance billing process to pay this year´s gas bill.
Balance billing is a billing practice, where provider bill you the difference between total cost and allowed amount. It can be paid partially or monthly. It generally in practice for medical billing as healthcare bill are generated for difference between total cost and amount which is paid through insurance.
Now, lets find out monthly bill paid by Maria.
We know there are 12 months in a year.
∴ Monthly payment= [tex]\frac{\$ 2388}{12} = \$ 199[/tex]
∴Maria will pay $199 monthly as her gas bill this year.
Maria's monthly payment for natural gas this year using balance billing will be $199.
Step 1
Balance billing is a method where a utility company estimates the annual energy usage and divides it into equal monthly payments. To find Maria's monthly payment for natural gas this year, we divide the total amount she spent last year by 12 months.
Total amount spent last year = $2,388
Step 2
Monthly payment this year = Total amount spent last year / 12 months
[tex]\[ \text{Monthly payment} = \frac{2388}{12} = 199 \][/tex]
Maria's monthly payment for natural gas this year, using balance billing, will be $199, calculated by dividing the total amount she spent last year by 12 months.
70 points!
Write the equation that represents the proportional relationship displayed in the table.
y = 4.5x
Explanation:
9 ÷ 2 = 4.5
18 ÷ 4 = 4.5
27 ÷ 6 = 4.5
The country of Freedonia has decided to reduce its carbon dioxide emission by 35% each year. This year
the country emitted 40 million tons of carbon dioxide.
Write a function that gives Freedonia's carbon dioxide emissions in million tons, E(t), t years from today.
Answer:
The quantity after t years 40 [tex](0.65)^{\textrm t}[/tex] millions tons
Step-by-step explanation:
Given as :
The rate of decrease of carbon dioxide each other = 35%
The quantity of carbon dioxide emitted this year = 40 million tons
Let the quantity of carbon dioxide emitted after t year = A millions tons
Now, According to question
The quantity of carbon dioxide emitted after t year = The quantity of carbon dioxide emitted this year × [tex](1-\dfrac{\textrm rate}{100})^{\textrm time}[/tex]
Or, A millions tons = 40 millions tons × [tex](1-\dfrac{\textrm 35}{100})^{\textrm t}[/tex]
Or, A millions tons = 40 millions tons × [tex](\dfrac{100-35}{100})^{\textrm t}[/tex]
Or, A millions tons = 40 millions tons × [tex](\dfrac{65}{100})^{\textrm t}[/tex]
Or, A millions tons = 40 millions tons × [tex](0.65)^{\textrm t}[/tex]
So,The quantity after t years = A = 40 [tex](0.65)^{\textrm t}[/tex] millions tons
Hence The quantity after t years 40 [tex](0.65)^{\textrm t}[/tex] millions tons Answer
Tim bought a book that was marked down to 50% of its original price. He used a coupon to save an additional 40% off of the sale price. If the book's original price was $12.00, what was the final price Tim paid?
Answer:
The answer would be $3.60
Step-by-step explanation:
50% of $12 is $6. Take 40% off of that and you get $3.60.
Answer:
The answer is 2.40
Step-by-step explanation: That is the answer because 50 percent into 12 is 6 and 40 percent into 6 is 2.4 = 2.40
Ramona spend 8 hours making picture frames if each picture frame takes her 1/4 of hour to make how many picture frame does Ramona have?
Eric has two dogs. He feeds each dog 250 grams of dry food each, twice a day. If he buys a 10-kilogram bag of dry food, how many days will the bag last
Answer:
10
Step-by-step explanation:
okay so if he has 2 dogs and he feeds each dog 250 grams twice a day
if he buys a 10 kilo bag, how many days will it last?
so first off, convert the 250 grams into kilos. which is 0.25
0.25 × 4 = 1 kilo
4 is the amount of times he feeds the dogs.
so in reality, it should take 10 days.
The number of days will the bag last is 10 days.
Given that,
Eric has two dogs. He feeds each dog 250 grams of dry food each, twice a day.Based on the above information, the calculation is as follows:
Here first we have to convert the 250 grams into kilos so it should be 0.25
And, there is no of times the dog feed is 4
So, it should be 1 kilo
And, there is the purchase of 10 kilos
Therefore, we can conclude that The number of days will the bag last is 10 days.
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Jeremy randomly selected a round tile and a square tile from the sets shown below. What is the probability that Jeremy
selected either two blue tiles or two white tiles?
a. 23/26
b. 4/9
c. 7/72
d. 1/40
Based on the number of square and round tiles, the probability of selecting two blue tiles and two white tiles is d. 1/40.
What is the probability of Jeremy's selection?The probability of Jeremy's selection can be found as:
= (Probability of picking round white tile x Probability of picking square white tile) + (Probability of picking round blue tile x Probability of picking square blue tile)
Solving gives:
= (2 / 36 x 9 / 60) + (6 / 36 x 6 / 60)
= 0.025
= 1 / 40
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Struggling un this section
Answer:
[tex]10[/tex]
Step-by-step explanation:
Combinations : When there are total [tex]n[/tex] choices and you have to choose [tex]k[/tex]of them (in any order)
The possible ways [tex]= \ ^n{c_{k}} = \frac{n!}{k!(n-k)!}[/tex]
Here total chances [tex]=5[/tex]
and we have to choose [tex]2[/tex]of them.
Possible ways[tex]=\ ^5{c_{2}}[/tex]
[tex]=&\frac{5!}{2!(5-2)!} = \frac{5!}{3! 2!} \\\\=&\frac{5 \times 4\times 3!}{3! 2!} = \frac{5 \times 4}{2} = 10[/tex]
an experimental model for suspension bridges built In one section, cable runs from the top of the tower down to the roaday, touching there, and up again ,to the top of the second tower. The towers are both 16 inches tall and stand 80 inches apart. At some point along the road from the lowest point of the cable, the cable is 1.44 inches above the roadway. Find tha distance between that point and the base of the nearest tower.
The distance between the point where the cable is 1.44 inches above the roadway and the base of the nearest tower is : [tex]\[\(28 \text{ inches}}\][/tex].
To find the distance between the point where the cable is 1.44 inches above the roadway and the base of the nearest tower, we need to understand the mathematical model for the suspension bridge cable. This cable typically forms a parabolic shape.
Let's establish our coordinate system:
- Place the origin [tex]\((0, 0)\)[/tex] at the lowest point of the cable.
- The towers are at [tex]\(x = -40\)[/tex] inches and [tex]\(x = 40\)[/tex] inches since they are 80 inches apart.
- The height of the towers is 16 inches.
The equation of the parabola can be written in the vertex form [tex]\(y = ax^2\)[/tex] since the vertex is at the origin [tex]\((0, 0)\)[/tex].
At the position of the towers (x = ±40 inches), the cable is 16 inches high:
[tex]\[y(40) = 16\][/tex]
[tex]\[a(40)^2 = 16\][/tex]
[tex]\[1600a = 16\][/tex]
[tex]\[a = \frac{16}{1600} = \frac{1}{100}\][/tex]
So, the equation of the parabola is:
[tex]\[y = \frac{1}{100}x^2\][/tex]
We need to find the point where the cable is 1.44 inches above the roadway:
[tex]\[\frac{1}{100}x^2 = 1.44\][/tex]
[tex]\[x^2 = 1.44 \times 100 = 144\][/tex]
[tex]\[x = \pm\sqrt{144} = \pm 12\][/tex]
Therefore, the points where the cable is 1.44 inches above the roadway are [tex]\(x = 12\)[/tex] inches and [tex]\(x = -12\)[/tex] inches.
The distance from [tex]\(x = 12\)[/tex] inches to the nearest tower at [tex]\(x = 40\)[/tex] inches is:
[tex]\[40 - 12 = 28 \text{ inches}\][/tex]
Similarly, the distance from [tex]\(x = -12\)[/tex] inches to the nearest tower at [tex]\(x = -40\)[/tex] inches is:
[tex]\[-12 - (-40) = 28 \text{ inches}\][/tex]
Thus, the distance between the point where the cable is 1.44 inches above the roadway and the base of the nearest tower is:
[tex]\[\boxed{28 \text{ inches}}\][/tex]