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3 years ago

15

For small changes in temperature, the formula for the expansion of a metal rod under a change in temperature is: g-L=aL(t-T), wh

ere g is the length of the object at temperature t,and L is the length at temperature T,and a is a constant which depends on the type of metal a) express g as the linear function of t.Find the slope and vertical intercept in terms of L,T,a

1 answer:

nasty-shy [4]3 years ago

3 0

Answer:

$g(t) = aLt - aLT + L$

$Slope = aL$

The vertical component is: $L - aLT$

<em></em>

Step-by-step explanation:

Given

$g - L = aL(t - T)$

Solving (a): Write g as a function of t

$g - L = aL(t - T)$

Open Bracket

$g - L = aLt - aLT$

Add L to both sides

$g - L + L= aLt - aLT + L$

$g= aLt - aLT + L$

Represent as s function

$g(t) = aLt - aLT + L$

Since the function is a function is t:

The slope is the coefficient of t;

Hence:

$Slope = aL$

To get the vertical intercept, we set t to 0

$g(0) = aL * 0 - aLT + L$

$g(0) = - aLT + L$

<em>Hence, the vertical component is: </em> $L - aLT$ <em></em>

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3 years ago

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3 years ago

At what point does the curve have maximum curvature? Y = 4ex (x, y) = what happens to the curvature as x → ∞? Κ(x) approaches as

MAXImum [283]

<u>Answer-</u>

At $x= \frac{1}{2304e^4-16e^2}$ the curve has maximum curvature.

<u>Solution-</u>

The formula for curvature =

$K(x)=\frac{{y}''}{(1+({y}')^2)^{\frac{3}{2}}}$

Here,

$y=4e^{x}$

Then,

${y}' = 4e^{x} \ and \ {y}''=4e^{x}$

Putting the values,

$K(x)=\frac{{4e^{x}}}{(1+(4e^{x})^2)^{\frac{3}{2}}} = \frac{{4e^{x}}}{(1+16e^{2x})^{\frac{3}{2}}}$

Now, in order to get the max curvature value, we have to calculate the first derivative of this function and then to get where its value is max, we have to equate it to 0.

${k}'(x) = \frac{(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})}{(1+16e^{2x} )^{2}}$

Now, equating this to 0

$(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x}) =0$

$\Rightarrow (1+16e^{2x})^{\frac{3}{2}}-(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})$

$\Rightarrow (1+16e^{2x})^{\frac{3}{2}}=(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})$

$\Rightarrow (1+16e^{2x})^{\frac{1}{2}}=48e^{2x}$

$\Rightarrow (1+16e^{2x})}=48^2e^{2x}=2304e^{2x}$

$\Rightarrow 2304e^{2x}-16e^{2x}-1=0$

Solving this eq,

we get $x= \frac{1}{2304e^4-16e^2}$

∴ At $x= \frac{1}{2304e^4-16e^2}$ the curvature is maximum.

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2 years ago

The sum of two numbers is seventeen. Three less than three times the smaller is equal to the larger

tia_tia [17]

The numbers are 5 and 12.
Smaller number = x
Larger number = y
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3 years ago

Please help if you know the answer

worty [1.4K]

It's C. When you're dividing g(x) by f(x), you get the equations in both B. and C., but C is more correct because you can't have x=-1/3 because the function is undefined at that number (you can't have the denominator equal 0)

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3 years ago

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